I have the same setup you do. I use the heater when I am in the shower. Once I have toweled off I turn off the heater and turn on the fan. If your bath fan is strong, it could exhaust the warm air before it ever heated the room if you use both of them at the same time.

I have the one in the link below. I'd say it extends about an inch from the ceiling, so not completely flush, but close. Mine appears nearly square, so I am unsure about the measurements I see online. I'd guess maybe 13 x 14-ish. If you want the heater, it may be larger. I opted not to do the heater because I installed floor heat and didn't think I would need something to blow warm air into the room while simultaneously sucking moist air out. Here is a link that might be useful: panasonic fan

Many of the fans with a heater have a small heating element with a second fan---the problem is that many of those models also require three electrical feeds to allow for independent use of all the features---light, exhaust fan, and heater. Most units with only fan and light have a 3 wire cable installed for operation. When installing the models with added heaters, an additional feed must also be installed.

Yes you can though sometimes it's a bit rough to pull an additional conductor through a conduit that already has stuff in it. They make double switches that fit where one switch goes. The questions are: 1. Can the circuit handle it. Most likely yes, the heater+light rarely exceeds 1200 Watts. Of course, you'll have to decide what else is on that circuit (not really so much a code factor than a nuisance factor). If it's on the same circuit as the receptacles in the bathroom, anybody with teenager daughters knows that one of those monster blow driers can trip a circuit. 2. Can you shoehorn the unit you want into the space your existing fan is in and get the exhaust duct to line up.