I can't see your picture right now, but basically yes - you need an H-bridge, either made with transistors, or made with relays (more or less limiting you to full on or full off tho).
There are several H-bridge controllers available, such as L293 or L298, do a litle searching for their equivalent if you need mroe current control (you need to take the stall current of the motors into account whens selecting your parts).
Not necessarily - if you stuck an NPN or n-channel mosfet between the ground rail and the "low side" relays (with an appropriate current rating - and a snubber diode), you could use PWM that way; of course, going this route, you might as well make the entire thing from transistors/fets to share the load (and for really large motors, using a single device for the PWM control would probably be almost impossible anyway).