Integrated circuits have a range of usable voltages and in this case it is most likely driven at 3.7v because that is what the voltage of the battery (atleast 3.7v sounds like a lithium battery), and not because that is the optimal value to drive this particular IC at. But what the H-bridge can and cannot tolerate is something you have to look up in a datasheet if you want to be sure - so if you can read the partnumber just use Google.
So if you don't wanna try to control it with 5 volts you have a few options.
Make a voltage divider to drop the voltage to 3.7v it only requires some resistors.
buy a 5v to 3.3v level shifter.
Insert 2 diodes or a red LED in series - this will drop the voltage to 3,8V and 3V respectively. You can always salvage a red LED somewhere if you have no Electronic components at hand.
Yes you can use pwm to control the motor speed. PWM is just a way of turning the power all the way on and off repeatedly and by that way modulating how much power the motor receives on average, thereby changing the speed. Your H bridge might even have a dedicated PWM pin making it easier for you- I assume thats what you mean by build to PWM Again you need the datasheet to know this. Otherwise if no PWM pin is present doesn't mean you cannot run it at PWM, just that you have to give PWM commands to all your input pins with the analog write command.