I like to think the opposite. I want my horse to seek release from the bit and be responsible for his own frame. Maybe it's just semantics, but when I hold, I want him to find that release and rely on the release. That is aided by the forward movement.
For colt starting, I usually do a lot of lateral flexing. Holding the rein to my leg, and when the horse gives his head, I release the pressure completely. I do this a thousand times until I can get that suppleness with a pinkie amount of strength. That way when you are moving forward, asking with both reins, the foundation for "seeking the release" has been put down.
For a freshly broke two-year old to retain learning on 1 to 2 rides a week - it's not going to happen. It'll be very slow going until you get your own place, and I wouldn't expect a lot out of the horse. I would have to be riding my 2 y/o every single day before I expect him to learn higher concepts such as holding a frame.
On the subject of if he is trotting and breaks into a lope, I would maybe let him lope a few strides, but I would bring him back down with a one-rein stop. You didn't cue him, and if he has a problem dealing with the pressure of your legs, he can deal with it going in a circle - looking for that release, yet again.