I found that the neutrals are tied together in this box, is this allowed?
Absolutely never. If a box contains 2 different groups of hot wires that do not interact with each other, their neutrals must not interact either.
Neutral is the normal return current path, it flows the same amount of current as the hot. You notice we don't put fuses on neutrals. Why should we have to? Neutral flows the exact same amount of current as hot, so the hot breaker will protect it... right?
Well, as soon as you cross neutrals between circuits, you wreck that protection. Now 1 neutral could return 2 hots' worth of power. Now we need fuses or circuit breakers on neutrals. No thanks.
Also, AC power throws quite a powerful magnetic field proportional to current. As long as currents are equal and opposite among wires in the same cable, those magnetic fields cancel each other out. However if current flows up 1 cable and back another cable, it creates all sorts of mischief.
Also, we do not like paralleling - providing 2 pathways for the same current. It causes many problems, including the way it fails silently and destructively if 1 of the 2 wires has a loose connection, forcing 2 wires worth of current onto the other wire.
Can grounds on these different circuits be tied together?
Yes, that is fine. Because grounds only carry current during a fault condition, which we hope is ended in seconds when the breaker trips.
I'm going to be replacing my breaker panel soon and would like to try and install AFCI breakers if possible. Would the tied grounds cause any interference?
xFCI breakers have zero tolerance for crossed neutrals. The xFCI will incessantly trip until you correct all of them.
However, they don't care about grounds. If you look very closely at a breaker and how it attaches to the panel, you see the breaker doesn't even have access to ground.
However, xFCI devices will seem to care about ground. That's because ground enhances their ability to detect ground faults. Diagnostic procedure will reveal that removing the ground stops the incessant trips, and so the novice will blame ground. In fact the appliance is faulty, and removing ground only defeats the detection. It's like pulling the battery out of a carbon monoxide detector because it won't shut up.
I'd like to get some smart switches inside this 2-gang box, but it's pretty tight in there.
Is there anything I should know before I put a 3gang box in there?
You don't need to. You can use a box extension.
One of the most elegant is the Legrand Wiremold Surface Conduit Starter Box, which gives about a 1" lift. It's meant for you to then attach surface conduit to the sides of it... but if you don't do that, well, I won't tell :)