Can I hire an opposing coach as long as I'm promoting him? If so, how do I go about that? I also have to hire a bunch of position coaches. I have to wait for Early Free Agency to do that, right?

As long the one you wanting to hire is not under current contract, hire away as long as its a promotion. You cannot hire an OC for an OC/DC position and a position coach for another position. You must hire him as a head coach for OC/DC and either a OC/DC as a position

During early free agency you can hire any coach from another team as long as it's a promotion. Outside of that, you can only hire coaches that are unemployed.