EDIT: In-dojo sparring is whatever Sensei says. Could be anything or nothing. If there is going to be much contact, at least hands and feet protection. Otherwise, it's up to you.
Tournaments:
We don't have an organization that spars officially, or a 'team' for tournaments. They're not required in our dojo. However, there are lots of open tournaments around the SE Michigan area, and I've also taken part in a number of open and strictly Isshin-Ryu tournaments in other states. All of them had the same requirements for point-sparring.
Padded headgear.
Open-finger hand guards.
Mouth protector.
Groin guard / cup for men.
Feet protectors.
Shin guards are optional. I wear them because I have a bone bruise on one shin that's over a year old - going on two years old - from when a guy blocked my kick with a kick and our shins smashed together. If people want to go through that, yay them; I don't.
Chest protectors are also optional; I see a lot of kids and some adults wearing them. Personally I don't own one. I've never taken a punch or kick to the chest or stomach in sparring that I could not handle.
Black belt sparring typically allows stikes to the face (light touch as they say). Underbelt sparring typically disallows direct face contact, but the side of the head or the top of the head is OK. Rules tend to vary and I've also noticed enforcement is a bit spotty. Some tournaments give 2 points for head shots, some only 1 point. The 2-point head shot tournaments makes it a head-hunter's game.
Generally no kicking to the back of the opponent, and no supporting-leg sweeps. Front leg sweeps are OK. I seldom see any sweeps at all, of any kind.
No throws, no 'blind' techniques; that is, no 'no look' reverse kicks or strikes. If you knock an opponent down, you generally have the count of two seconds to deliver a simulated technique (no hard punching to a downed opponent, just pretend you did it) and you get the point. If you don't deliver a simulated technique, no point for knocking him down.
I know of one dojo that requires all their karateka to wear groin protection; even females. And it's not a 'cup', it's a full-on wrap-around hard plastic thing that covers from top of groin to rear, uh, ventral fin. Must be very uncomfortable. That's just a dojo thing, though; I've never seen a tournament require it.
I also wear my prescription eye goggles under my helmet. I haven't yet been told I could not have them, and I need them to see.
I have also seen a couple of tournaments require different gloves for continuous sparring; closed-hand. I haven't tried that yet and don't have any closed-hand gloves anyway. Most tournaments I've been to do not require closed-hand gloves for continuous sparring though, so that may be a new thing.
Most tournaments that I have been to also state they require all competitors to wear a clean traditional karate gi; but many show up in sweat pants and t-shirts with a sash or belt tied around the waist and they're allowed to compete anyway. Judges are required to show up in a gi; many do not. Again, rule is commonly not enforced. Personally, I think that takes away from the tournament's class and prestige. If it's a karate tournament, wear a gi like a karateka or stay home.