I am kind of a beginner (and a self-learner) on the piccolo trumpet. I have seen people playing the instrument's fourth valve with their right hand little finger, and others with their left hand index. Is there a reason to choose one over the other, is there a configuration which definitely provides more endurance or agility in the long term?
Depends on the way the instrument is constructed. Most piccolo trumpets, especially ones with piston valves, are designed for the valve to be played with the fourth finger of the right hand:
You can see that the 4th valve is even offset a hair to make it easier to reach.
Some rotary valve piccolos have extra spatulas that are operated with the left hand:
Picture from http://www.trevorjonesltd.co.uk
I'm having trouble finding a picture of a 4-valve picc with the 4th valve played with the left hand, but I know I've seen it. It's the same principle as a euphonium's fourth valve:
The thing about piccolo trumpets is that they're so small and the tubing is so tightly wrapped that there's often no comfortable way to hold them. So while I'm normally a stickler for holding an instrument properly, I think there's nothing wrong with operating a picc's 4th piston valve with the left hand. I say let it be a personal choice.
Not being a trumpet player but a guitarist, I could be wrong however, I'd imagine people use their pinky finger to build up strength in that finger, as usually your small finger will be your weakest (the same is true for guitar, people do exercises to build up finger strength the same way).
I'd imagine it's also a comfort thing, so unless an actual trumpet player can correct me, I'd say try both and see what's best for you but remember to build up your finger strength regardless of how you play!
The first three valves work just as a normal trumpet, and will be pressed with index, middle and ring fingers. The fourth valve is generally to produce a pitch a fourth below, so doesn't get used as much, although you may find tuning (intonation) better by using that in combination with other valves. It's somewhat like pressing valves one and two gives a note like just pressing valve three, but sometimes one combination is more in tune than the other. Either way works, but I feel that using only one hand will be better in the long run, while the left steadies the instrument. You may find after a while that you can use either, so keep your options open!