Mulberry Help

I live in the Phoenix, Arizona area. About six months ago, I noticed what I thought was a weed growing between my backyard shed and patio. Since I don’t have any other plants back there, I decided to just let it grow and see what happened. As it kept getting taller, I got curious and did some research. Turns out, it’s very likely a mulberry tree!

I’m really excited about the idea of having some shade or even fruit in the backyard.

The spot it’s in isn’t ideal, though. There’s not enough space for it to grow properly, so I plan to transplant it to the center of the yard at some point. From what I’ve read, it’s best to do that in winter while it’s dormant, so I’ll probably wait until November or December, unless anyone thinks it’s a good idea to do it now?

In the meantime, I’d like to train it as a single-trunk tree. Should I go ahead and prune the low branches near the base now, before it warms up too much? Or would it be better to wait until the time I transplant it?

Also should I stake the tree? It has a very minor lean. Thank you!