Just use your common sense. Everyone has different luck with gloves, I generally don't use them on fine work, because it interferes with my hands. Those points easily penetrate gloves anyway. The straight needle tipped springs, just be careful around them.
If you have never disassembled saxophone keywork, before you take a dive into purchasing pads and the accessories, if you have a jeweler's screwdriver set, you can try removing a screw or two, then put it back in.
This will also help you to assess whether such screws are corroded to the extent where they can or cannot be removed. Above all, use the proper width jeweler screwdriver that is the width of the screw head slot, to prevent from damaging the screw head slot surfaces. In general, the hinge screws are of a softer steel, IMO like grade 1 or 2, not hardened like grade 5, or grade 8 used in automotive engines.
Because the keys are interwoven amongst each other in certain places, I've had to remove up to say, a half dozen hinge screws. It is not difficult but tedious. There should be no reason to bent key/pad work to remove or get things to fit.
Every person differs in their ability to take stuff apart and then get it properly put back together again. Getting the key/pad work back together again, in some cases requires considerable patience. I've had to even take a breather and step away, tackle it when I am ready, again.
(Pet and feed the cat, water plants, check the mail, do a little gardening, go shopping, call someone, etc. I no longer have pets, but I think you know what I mean. And, your cat has a rather serious look on its face through its cute hazel green eyes. -
Do you really want to do this? Rub my tummy and I'll let you know. )
Not saying do not try, but like I said, a little experimentation first to get familiar would be helpful before you take the dive. Then you'll know if it would be more prudent to seek a skilled technician for this work.