I think in general, obviously anything that has to do with your image rights, or performance of sorts, will need to be done through your company. if you do it privately behind their back, then obviously you have breached your contract.
as for things like AMA or coming to lihkg, think that really depends on the company policy, management team, or just the artiste themselves. obviously if you are a major boy band, then alot of things will be at stake and will require more layers of approval, but for me im pretty chill with my company so they just pretty much let me do whatever.
but obviously this is not because of no reason, its simply because I've worked with them long enough, tight enough, and have proven myself enough times that I know what Im doing, and specially with positive feedback from you guys, that definitely makes things easier.
lastly regarding your question, alot of times when an artiste signs a contract there's some sort of advance, or allowance. some might be large some might not. so it really depends on your company.
Most of the time an artiste will probably choose not to do part time outside because there's some sort of "pride" or they want to focus more on the craft. of course if the situation gets to a point where they really cant make a living, I'm sure the company will also allow the artiste to go do something else, because cmon, that means the company has failed too innit.