Mustard gas is certainly lethal to rats, not to mention other small critters that inhabited the hellish landscape of the front. Plenty of data exists on exactly what mustard gas, not to mention just about any other hazardous chemical, will do to a rat as they of course are often the laboratory subjects for tests to determine exactly that, not to mention countless other unfortunate creatures like mice, pigs, dogs, horses, and so on. The super short summary though if we cut to the chase for the LD50 of mustard gas, as that was the one you mentioned (although I presume you are interested more in WWI chemical warfare than mustard gas, so will also speak more generally about the former's impact), it is as follows:

Species	Route	LD50 (mg/kg)
Human	Oral	0.7
Human	Dermal	100
Rat	Oral	17
Rat	Dermal	5

Other gases too, of course, were tested on rats and other animals, both during and long after the war. In a report from 1917 by the British Chemical Warfare Medical Committee, we see several examples of this:

Field experiments have shown that a rat and rabbit placed about five yards from, and to leeward of, a bomb containing phosgene were found dead within eight minutes after its explosion. Control animals similarly situated to windward died after several hours. Other controls, in which shells filled with water instead of toxic substances were used, have shown that the concussion must not be regarded as a cause of death.

Similar results have been obtained in laboratory experiments at Cambridge and elsewhere. Mice exposed to a high concentration of phosgene in a small closed chamber may die during the third minute.

The rats knew the danger the gas posed too. One of the warnings of an incoming attack for the soldiers in the trenches would be the fuzzy stampede as the rats which had been inhabiting No Mans Land fled in front of the oncoming clouds that drifted towards the trench, their squeaks of terror amplifying the official alarms. The aftermath would see the carcasses of rats strewn about, as well as other small mammals and avian life that had been eking out a survival there, or was forced into the trench of course, such as horses of dogs. A British study done in the wake of the war noted of one German attack carried out in early 1916, believed to be a mix of chlorine and phosgene:

Grass and other vegetation were turned yellow by the gas as far back as 1 ,200 yds. from the front line. Rats were killed in the trenches in large numbers. Eleven cows, twenty-three calves, one horse, one pig, and fifteen hens were killed in the fields behind the lines by gas, and a number of other cattle and pigs showed signs of being affected by the gas.

Speaking of a different attack, a British officer described the image of the rats much more vividly when he wrote of what a gas attack did to them:

The trenches swarmed with rats, big rats, small rats, grey rats, tall rats in every stage of gas poisoning! Some were sucrrying along scarcely affected while others were slowly dragging themselves about trying to find a corner in which to die. A most horrid sight - but very good riddance.

Good riddance indeed, as the rats were considered quite a scourge of the trench, and soldiers found that one of the few things that had any impact on their presence was the enemy, whether gas as above, or the explosions of artillery or grenades sending them scurrying away. It wasn't exactly a good trade-off, all in all though, since as made clear enough, even a gas attack could hardly be counted on to result in total lethality, and the ever common presence of heavenly conditions was sure to bring new rats in short order, likely a large contribution to literally millions of causalities in the war from the diseases they so easily spread.

After the war, men who had gained experience in chemical warfare in the war likely remembered the devastation that their weaponry had wrought on the local fauna, and in the 1920s attempts were made to 'pacify' gas warfare and introduce it as the newest way to protect crops from vermin and pests. The US Chemical Warfare Service conducted several tests in conjunction with the government on the promise that it offered "the quickest and surest method in attacking crop destroying pests, whether ground squirrels, gopher, blackbird, crows, buzzard, rats, or grasshopper".

Tests specifically aimed at rats were conducted in 1920 in the Gulf, in hopes of containing bubonic plague, and early reports claimed to be a rousing success with the use of a mixture of phosgene and chlorine gas. But testing in a contained 900 sq. foot area isn't the same as wide application, and the transition never was successfully made. Overall, such attempts to reapply chemical warfare to the civilian world showed that results were at best contained to the test conditions or short-lived, and in some cases quite counterproductive.

One attempt at tent caterpillar extermination found that it was killing the plants too. Other gasses were found to be ineffective, and the ones that worked best were of course the ones that were worst for humans as well. And the most desired target, the boll weevil, proved to be impervious to whatever was thrown its way as it simply could survive without breathing. Nevertheless it did help lay the groundwork for research that would eventually lead to more successful pesticides, even if not from the Chemical Warfare Service.

Now, as for lice, I would on the whole have to defer to someone else to note if there was any impact. I can make some guesses, but I wouldn't want to speculate. I would simply note that there is some very promising titled material in the National Archives which seem to be undigitized in the Records of the Bureau of Entomology and Plant Quarantine, especially "Report on Experiments Conducted on October 16, 1918, Testing the Effect of Certain Toxic Gases on Body Lice and Their Eggs" in Record Group 7.

Sources

Brantz, Dorothee. "Environments of Death" in War and the Environment: Military Destruction in the Modern Age, edited by Charles E. Closmann. Texas A&M University Press, 2009.

Cook, Tim. No Place to Run: The Canadian Corps and Gas Warfare in the First World War. UBC Press, 2011.

Faith, Thomas. Behind the Gas Mask: The U.S. Chemical Warfare Service in War and Peace. University of Illinois Press, 2014.

Gupta, Ramesh. Veterinary Toxicology: Basic and Clinical Principles. Elsevier, 2011.

MacPherson, W.G., Herringham, W.P., Elliott, T.R., & Balfour, A. eds. Medical Services: Diseases of the War, Vol. II. His Majesty's Stationary Office, 1923.

Medical Research Committee, Reports of the Chemical Warfare Medical Committee of the Medical Research Committee: No. 1, Physiology (War) Committee of the Royal Society. 1918.

Russell, Edmund. War and Nature: Fighting Humans and Insects with Chemicals from World War I to Silent Spring. Cambridge University Press, 2001.

“War on Rats with Poison Gas.” Chemical and Metallurgical Engineering 23, no. 11 (September 15, 1920): 543.

Young, Leslie. "Observations on the Effects of Mustard Gas on the Rat". Canadian Journal of Research, 1947, 25e (3), 141-151

ETA: A few more quotations to add a bit more color.

Also, don't miss /u/voidoid's piece here in the thread which is a nice complement.

Mustard gas, chlorine, and phosgene were three of the most common gases used. Note that while chlorine and phosgene do have the suffocation effect you mention, mustard gas is actually a blistering agent - and it's actually not even a gas, although inhaling the vapors could cause pulmonary edema in addition to first and second degree burns...the effects of mustard gas, in other words, were not suffocation per se.

With regard to the effects of chemical warfare on the rats and lice in the trenches, there's some anecdotes among combat memoires that suggest an overall limited or unnoticeable effect on population (again - this is anecdotal). For example, in With a Machine Gun to Cambrai (I use this example in particular because coincidentally, Cambrai marked the first use of a mustard agent by the Allies), George Coppard, serving with the British across several major battles in France during 1916-1917, noted that "What happened to the rats under heavy shell-fire was a mystery, but their powers of survival kept place with each new weapon, including poison gas." It should be noted that the day in, day out dealing with rats could have an effect on the anecdotal observations of soldiers who may have simply not noticed a slight reduction in rat population after a gas attack.

With regard to lice, one thing to bear in mind is that any protective equipment meant to entirely cover the body and clothing would have protected the lice as well. This situation applies to a mustard agent, which in part is designed to soak through clothing, blistering and burning the skin and ensuring the victim would inhale noxious vapor from the chemical soaked into his clothes. So...if you protect the clothing, skin and hair, you're protecting the home of the lice. I can speak to the rat issue more than lice, though.

Realize that the purpose of mustard gas was not to kill, but incapacitate. Only 1% of exposure cases in WW1 were lethal. But mustard agent settles (heavier than air) and sticks around (for weeks), and continues to cause bad effects if one comes into contact with it after the initial attack. In fact, as the war progressed, higher and higher concentrations of mustard agent were used in combination with aerosols (since, again, mustard agent is not a gas) to cause soldiers to abandon certain areas of line - however, these would not be reoccupied by the attackers...this is an example of area denial. It's very likely that in the areas that were heavily attacked, whether or not the rat population was affected would be unnoticed by soldiers since they'd have evacuated that section of line, and rerouted accordingly. Their new space would look particularly inviting to the rats that evacuated with them.

Okay, so what about rats that stuck around after an attack?

With regard to a nominal amount of mustard gas in the environment, a 1975 study found that "No deaths attributable to sulfur mustard were noted in mice, rats, guinea pigs, rabbits, or dogs exposed to 0.1 mg/m3 (0.015 ppm) of sulfur mustard vapor, 6.5 hours/day, 5 days/week, for up to 1 year (McNamara et al. 1975)." So simply being around the trenches after most of the chemical dissipated had likely no effect on the rat population. The same study found a small likelihood of chronic keratitis in that population, which would affect rat vision. This study also found some effects in birth rate and mutations, but this is essentially null considering the over all birth rate for rats in the trenches (producing upwards of 900 offspring per year per couple). Again, we're only talking about the impact of low levels of exposure over time.

The oral LD50 for rats is 17 mg/kg (LD50 is the median lethal dose). That's what the concentration would need to be if, for example, a rat were to chomp down on a blown off decomposing hand in a trench. Acute exposure (environmental) to 2.5mg per day could take 10 days to death, though...so this should give a bit of a range in understanding how much a rat could take before dying. Also, up to 15.5mg could be applied to the skin according to a 1994 study with only moderate blistering. What does this mean? A rat can withstand a fair amount of environmental exposure that would not affect breeding, but direct ingestion would be more likely to cause death.

I'm not going to go much further into the various effects, but here's a CDC source for further research: link.

I've taken more of a deep dive into mustard agent, but note that phosgene was the primary killer (85%) of gases used in WWI. Chlorine killed much faster, but phosgene was used primarily. Both would have a severe effect on rats, would would not be able to withstand anywhere close to an LD50 for humans for either gas. In the German attacks on British troops at Hulluch in April 1916, using a mixture of both phosgene and chlorine, rats were seen fleeing German trenches prior to the attack - the assumption was that this was due to leaky cylinders (the Germans had stored nearly 7,500 cylinders along the line to produce two major gas cloud attacks).

Bottom line - yes, rats could be affected by all three of the major gases (I'm excluding tear gas, which was used more frequently earlier in the war but is clearly not meant to kill or deny area access). However, the population would not likely be locally affected enough for a noticeable difference by soldiers.

The one reason a soldier would notice an immediate dissipation of the local rat population would be during an aerial bombardment, in which rats always seemed to disappear. This is by no means anecdotal, and was observed on a widespread basis throughout the war.