Most likely not. My guess is that the air conditioner is overloading the circuit. You probably need to have a dedicated circuit installed to power the air conditioner. A licensed electrician can easily help you with this.

You will need to have a 30 amp line installed. You cannot change the outlet and breaker alone, nor the cord on the unit - these actions are both potential fire hazards.

The fan needs to be vented somewhere to prevent mold and mildew buildup. If it’s vented into an unheated space such as an attic, then that would be OK.

No. If the heaters are off, then they’re not drawing any power.
It's hard to say if they are 'hard' on your utility bill, because I don't know what this is based on. Electric baseboard heat is the most efficient type of heat available, but if your electric rate is more expensive than propane or oil, then it may cost more to heat your home with it.

I have a 12’ x 12’ sunroom that I want to heat with electric baseboard. How many do I need to install?

There are many factors that go into determining heating size. Basically, it requires that you do a heat loss calculation for the room. This takes into account factors such as number of windows and direction they face, number of inside and outside walls, insulation in the walls, and so on.

Here's a good article that should be able to help you calculate heating needs