Hello there! I'm definitely not an expert on this topic, but I can make a few comments on each of your questions:

(1) The state would presumably have provided the praetorians their arms and armor, though the expenses would also be deducted from their pay. But in Rome, where they spent most of their time, the praetorians simply wore togas (Tacitus calls them the cohors togata, Histories 1.38; see also Annals 16.27). Roman soldiers generally had to pay for their own clothing as well.

(2) I should point out that it is somewhat anachronistic to describe ancient soldiers as wearing "uniforms," since equipment would have varied depending on what individual soldiers procured from their quartermasters, private armorers, or their own families (in a rather extreme case, a soldier named Terentianus stationed in Egypt asked his father by letter to send new boots, socks, cloaks, tunics, and even tools and weapons!). The ancient Romans, as far as I'm aware, attached no special significance to uniformity. But as I mentioned above, praetorians normally wore togas, and that would have lent a unique appearance. This is perhaps confirmed by the historian Herodian, who describes how a deserter named Maternus attempted to assassinate Emperor Commodus during a festival by disguising himself and his fellow conspirators as "spear-bearers" (δορυφόροι), or otherwise praetorians (1.10.6). Maternus, by the way, was betrayed, captured, and beheaded before he could carry out his plot.

Otherwise, our limited evidence suggests that, at any given time, field equipment was similar to that of other legionaries. During the Battle of Bedriacum in the civil war of A.D. 69, Tacitus mentions how two praetorians, "having been disguised with shields taken from the dead," were able to approach the enemy's positions and disable a catapult before being killed (Histories 3.23). This would imply that only their shield blazons, whatever that was, distinguished them from other legionaries.

(3) From memory, various monuments seem to depict praetorians, though they don't look any different from their legionary counterparts. I would upload a picture, but I can't seem to find one at the moment amid the thousands of photographs I've taken at various museums. :(

Edit: Found something! Here's a tiny figurine of a Roman soldier (sorry for the poor quality) from the British Museum. The description reads: "Bronze statuette of a soldier wearing laminated armour (lorica segmentata) Roman, 2nd century AD--The soldier is usually identified as a legionary. He may be a praetorian guard (emperor's bodyguard), since similarly equipped soldiers are shown near the emperor, for example on the carvings on Trajan's column in Rome."