I've had all of my tetanus shots when younger with no serious side effects. Can I have a serious side effect if I get a booster shot as an adult?

Tetnus shot. You should have no problem getting a booster tetanus shot.
Very rare. but anything is possible. I would not worry about it.
Not likely. Tetanus boosters are recommended every 10 yrs and should be taken to prevent the deadly disease in case of accidental exposure with injuries etc. esp if traveling outside where it is more common.
Tetanus. Highly unlikely. Swelling and soreness are the likeliest side effects you would get if any.
Not likely. Although it is always possible to have a reaction to any vaccine at any given time, it is unlikely to happen if you are having a booster of sorts. Keep in mind there can be a reaction to any component of the vaccine.