You were given a very small influence on where you ended up within the army during the days of volunteerism. Initially when you volunteered you were allowed to pick a preferred regiment but this changed over time as the huge turnover in manpower meant that the smaller county regiments couldn't fill out all their battalions. The last men who were allowed to do this were the men who signed up under the Derby Scheme (https://en.wikipedia.org/wiki/Derby_Scheme). Following the introduction of conscription in early 1916, you were allocated to whatever regiment needed men at the time regardless of your geographic location or preferences. Your role in your regiment/battalion was decided by them. Thus during the dying days of volunteerism a certain amount of men signed up in order to be able to serve together and fight for the local county regiment rather than a random regiment they had no connection to. However this wasn't a statistically huge amount of men, as those who held passionate views of wanting to serve together had likely already signed up, especially via the Pals Battalions in the larger cities.

Even during the time periods when you were given a preference as to which regiment to sign up with you'd often find men posted all over the place, contrary to the choices they had indicated. For instance during the opening of the war in '14 there was a rush of men to the colours in Birmingham and the local regiments couldn't cope, so you saw large batches of men transferring elsewhere, especially to smaller county regiments who couldn't fill out their recruitment quotas locally. One example is the 9th Devonshires who were largely Brummie (i.e in the hundreds of men) with some South Welsh mixed in and only 20 native Devonians when the battalion was first constituted. This kind of recruitment was repeated all over the country.

Another example of soldiers being pushed towards certain services rather than choosing where they wanted to serve was in the recruitment officers 'suggesting' things to them. If a particular strapping athletic 6' 2'' man walked into an office and asked to sign up for the Devonshire Regiment, they'd often be pushed towards the Guard regiments, told about how famous these regiments were, how much of an honour it was to serve for them and persuaded to sign up for them instead. Likewise if a recruiter found a rather academically able volunteer they might suggest they work in an admin role instead, I came across a soldier in my own personal research who was offered an admin role when he went to sign up but refused this and ended up in one of the Guard regiments (Norman D. Cliff - To Hell and Back With The Guards). If you were particularly academically able you might decide to attempt this in order to avoid the trenches however it's important to note this was hardly full proof, literally hundreds of Oxbridge graduates died fighting in the trenches during the war in a period when the annual graduation rate was very small compared to today.

If you walked into a recruitment office, you were there volunteering to fight, I've not come across someone being able to ask to be a cook etc as that's not that you were volunteering for. However men who appealed against their conscription in 1916 onwards on conscientious objection grounds were often allocated such non combat roles in order to keep them supporting the war effort without having to fight. So technically speaking you could refuse to volunteer, refuse your conscription, go to the service tribunal and plead your case and end up not fighting if you're lucky and the tribunal believed your objection but this would have a massive impact on your social standing given Victorian social ideas. Most men didn't do this because they didn't want to be viewed as cowards and risk their social standing. Of those that actually did go this far and didn't just ask for a temporary reprieve, the most common from my experience were those with borderline mental issues and religious people such as Quakers and Salvation Army members, not people faking it to get a cushy support job.

So to answer the question, as far as I'm aware no you couldn't volunteer for a 'safe' job within the forces in order to escape the trenches, you could have some influence over where you ended up and sometimes through sheer luck or something interesting on your CV you might be able to avoid fighting in the trenches but you couldn't volunteer to be a cook, that wasn't the idea of the system and it would have been open to loopholes like you've identified. Of course you could try to find employment in a protected industry as /u/wafflefanny has explained but this is slightly different than trying to find a non combat military role.

As for whether the volunteers were better off in terms of being put in front of the German guns, not at all, they were thrown into dangerous situations just as much as anyone else. Indeed in '16 there was a certain contempt of conscripted men amongst the British Army hierarchy with some of the most egregious orders on the 1st Day of the Somme coming from generals who thought the conscripted men were unable to follow complex orders and do things like run across the battlefield while keeping formation/following plans, thus they in some cases preferred to use volunteers as they considered them better soldiers. Which would obviously put those volunteers in dangerous situations, not to mention the previous year's worth of fighting, which thousands of volunteers had died in since they were first deployed in large numbers at the Battle of Loos.

Source: CT Atkinson: The Devonshire Regiment 1914-1918 (in regards to the constitution of the 9th Devons), Peter Simkins: Kitchener's Army: The Raising of the New Armies: 1914-1916

Follow-up question: could you alternately avoid conscription altogether by getting yourself a civilian job necessary for the war effort, like working in a steel mill or operating a locomotive?

Could you volunteer for the navy? Could they still transfer you to the army? Statistically was the navy less deadly?

I want to add a piggyback question on this one, If I were a conscientious objector in England during this time, would I face any penalties?