Yes, pulling the string on the opposite hand will certainly speed up the yoyo. For example, throw a trapeze and as the yoyo is coming around the non throw hand give the throw hand a quick pull and release. You’ll find that that throw hand tug will send the yoyo across both hands into the position for a 1 1/2 mount. You don’t actually have to swing the yoyo to a 1 1/5 mount, you can just add that tug in the right spot to redirect the energy.
You’re really just shortening the arc of the loop around your hand to speed up the arc and then releasing it at the right time to send the yoyo in the right direction.
As for getting faster (and I’m not a very fast player myself so I, like you, are still learning) I’ve found that the first step is to learn a trick to the point where you’re not thinking in steps at all. Your hands just want to do the moves on their own. Also, get rid of transitions. What I mean by this is that we start to think of a trick as : step 1, step 2, step 3, and so on. Then we start doing steps 1,2,3 and 4 as a group, and then there’s the 2nd group of steps. There’s a transition between those two groups of moves. If you can start thinking of the moves in larger and larger increments then those transitions go away making the trick smoother and all of that lends itself to faster play.