"How long does the tank usually take to heat up?"
My electric heater takes 10 minutes for the water to feel warm. It takes longer to get hot.
My gas heater takes 5 minutes to feel warm and 15 minutes to get hot.
When both are on everything happens faster.
Note: Electric element will burn out if it is on and there is no water in the tank. It is a good idea to shut the heater circuit breaker off before you drain the hot water heater and leave it off until you put water into it again.
"I also just noticed that there is an unsealed gap around the base of the shower insert. Is that ok? Or should I seal it with silicone??"
If the gap is between the floor and the shower/tub pan, it is okay. You can install trim there if it bothers you.
If the gap is between the shower wall liner and the shower/tub pan there may be an issue. If leaking into the wall can occur, the shower wall liner needs to be reinstalled to close the gap.
Sometimes the shower/tub pan is not supported properly underneath and it can move when you step on it. That would also need to be corrected.
I wish you good luck and happy trails ahead!