1. So depending on if you are asking if they had "special forces" or simply "elite" forces there are varying answers. The First Cohort which was the most massive of all the cohorts* was made up of many of the best soldiers Rome had to offer and was the first in fighting force that had a massive reputation throughout the legion for being (for lack of a better word) tough ass guys (very much like the marines, except less aquatic). The Praetorians as well could be viewed as an "elite" force simply for the fact that they were powerful, well-known, influential military forces that also were heavily trained (and valued higher than any other military forces, at least in how they were paid). If however one was to try to pick a specific unit that might seem more "special forces"-like (although considering this is the ancient world it isn't going to be as rambo-ish) the Batavian Cohorts may meet the standards you were looking for. Having been used to attack the druids by undergoing an amphibious assault by swimming with their horses to attack (an impressive feat in that time period) this cohort specifically underwent sieges that typically legions couldn't. They weren't Navy SEALs by any stretch, but their aquatic abilities nevertheless were appreciated by Rome. The only unit I can think of that may meet both titles of "elite" and "special" was Legio X Equestris (10th legion mounted) under Julius Caesar. It was arguably his most trusted unit and fought under him in essentially every battle of the Gallic wars. In 45 BC they were disbanded but Mark Antony brought them back and used them as well (even going as far as to mint a coin specifically to honor them, which is a pretty big deal, coins are pretty sweet) they would eventually be merged into Legio X Gemina, and of all the units that come to mind exemplify the idea of a well known and tactically useful unit of Rome. Which for that time period is as "elite" or "special" forces as it gets.

2. Considering that the Praetorian Guard was made up of Roman Soldiers and were seen as being Roman Soldiers, it's hard to say if they were viewed differently as in separate from them. It'd be like looking at a different rank or a different unit in the armed forces. Rivalries exist sure, but at the end of the day they were all Roman. The Emperors view of the Praetorian Guard is what changed. You have Augustus Create them, then you see Caligula get killed by them, and Claudius become the new Emperor. The paranoia about them would rear it's face in many Emperors like Vitellius who disbanded them and recreated his own. So of course the view changed, but it changed (at least most visibly) with the Emperors.

6)Speculatores and Exploratores were reconnaissance/scouting units used by the legions. An Exploratore was a horseback unit that would ride ahead of the legion and scout the battlefield, returning to then provide tactical advice. A Speculatore is most likely what you were looking for, they would infiltrate enemy camps or cities, disguised as everything from a deserter to a merchant, and would and would proceed to gather information (or sometimes sabotage the enemy). Sometimes though leaders would simply go themselves as a "diplomatic visit" which would allow them to garner the information they needed on things like fortifications, etc.

Sorry I couldn't answer all the questions immediately, my knowledge on the inner-workings of the Praetorian Guard is rusty and I also don't have the time, I'll try to edit this and answer them later, as well as any follow up questions if you have any. I hope this helped.

3/4) Depends which period you're talking about. Initially Praetorian soldiers had to be Italian, so an Italian would join the Praetorians like they joined any legion. But after Serverus seized power he disbanded the Praetorians due to their scheming and refounded it as an elite unite made up of the best soldiers in the empire. Exactly how this worked, I'm not exactly sure. Severus's reforms stayed until Diocletian came to power and he eventually dissolved the unit. But surprisingly the position of "Praetorian Prefect" stayed and basically became a "Vice Emperor".

5. Romans definitely found infantry to be more honorable. Romans were never really good at cavalry and often resorted to hiring mercenaries. These attitudes could be seen in Procopius's accounts of Belasarius's battles. Procopius went to great length to glorify the infantry even though it was the cavalry that was decisively winning the battles.

The elite troops of any army have generally been the guard units and Rome is no exception. Depending on the time period, the Praetorians, the Scholae Palatinae, the Jovians and Herculians, the Excubitors, et al. have fulfilled that function.

The Scholae Palatinae and the Excubitors are notable examples, as they survived the transition to the theme system and remained in the tagmata until the 11th century. The Jovians and the Herculians were formed by Diocletian, possibly from the legions of his home Illyricum (Legio V Iovia and VI Herculia) and existed until suppression in the 7th century as elite field troops. Most of the guard units from Constantine onward were largely heavy cavalry - either clibanarii or cataphractarii.

Not all were regular army units. The emperors had used Germans, Goths, Magyars (among others) for centuries, but perhaps the most well known was the Varangian Guard. At first comprised of Scandinavians and Rus, eventually Anglo-Saxons dispossessed by the Normans, they primarily wielded large axes and fought on foot, and were known for their ferocity as berserkers.