Most Helpful Guy
It's best to play a little hard to get or guys (or girls for that matter) really will assume you're desperate. It's something everyone does with strangers, though they don't like to admit it.
It's kind of a rule of the thumb not to ask twice in a row. So yes, wait for him to ask.
Most Helpful Girl
A rule of thumb of mine is never send more than 3 unreplied messages. 3 is the absolute maximum.
I mention this because I think the basis of the rule should be applied here. You've put the effort in... But only continue to give it when it's been returned. Definitely wait until he asks. Guys DO like to do the chasing, and it is important that you show that you're interested. You've done that, and now let him come to you..
In terms of initiating conversation it should definitely be mixed. It's not fair for all the responsibility of starting texts to lie on one person. But do it less often than him.