Why not, it can pull warm air into the room.
It depends. If your bedroom has a supply AND return duct, then no. If there is only a supply duct, and you shut your bedroom door, this pressurizes the bedroom and limits the CFM delivered into the bedroom. A room to room vent or duct would equalize this pressure and allow maximum CFM to be delivered from the central air/heat system.
depends on room size
Typically not, a vent free should not be installed in an enclosed area, ie. bedroom, bathroom. They do make "bedroom" sets that are a very low BTU. The flame is very small leading most to chose another option.
The air from the register/vent blowing into a bedroom must find its way back to the central cold air return somehow. This is usually accomplished by having a large enough crack/cut under the door above the carpet or flooring for the air to move through. If you do not allow the bedroom air to flow back to the central cold air return, the bedroom will become uncomfortable - usually to hot or to cold. Another solution is to install a grille in the bedroom door to allow the air to flow through.
There is a little knob in the girls room that controls the vent
Closed dampers or registers, not sure what you mean by vent. Chimney? Return air grille?
The air from the vent should be 18 to 22 degrees F lower than the temperature of the air going into the return.
Clear the vent tube
I would say ok on the supply but not hte return.
you push the vent
It depends on whether you have a mount for the microwave that has an open bottom or if there is a cabinet over the stove. If there is only a mount then yes there are special over stove micros that bottoms are heat resistant to the stoves heat. If there is a cabinet then you can use a regular micro. Either way there should be a vent from the compartment that the micro is in or it could over heat. The special micros have a built in vent that can double as the stove vent.
Then you can go back and turn on the heat vent :)