If I wanted to connect let's say 4 motors in h bridge configuration while using the same power supply (12v) for all 4 of them and would also like each motor to be able to work independently, what do I need to do? Is this feasible (without a motor driver IC)? Would hooking up more than one h bridge to the same power source affect the current that goes through each motor? If so, how to remedy that?
If you want 4 independent motors then you need four independent H-Bridges.
If you have a 12V source that can supply current to all 4 simultaneously then there's nothing to worry about. Hook them up in parallel and get a controlling circuit (MCU, timer, etc.)
If your 12V supply can't provide enough current to drive all 4 motors then you need to ask more questions:
3a. Do you allow default coasting (non-tracking state)? If yes, you can use that state to save power and then simply apply power to each of the motors out of phase with one and other (i.e. if you can drive two motors at a time, then only drive two at a time and have the other two coasting).
3b. If you don't allow coasting then the solution is get a bigger power supply.