Hi, Aaron. You've basically got it right. Since the Tempest-X has two 8 ohm voice coils, you need to wire them together (in parallel--red to red, and black to black) to knock the impedance down to 4 ohms for the O-Audio plate amp.
No, you don't need to solder the speaker terminals. When you push the connectors in, they expose a rather large hole into which you can insert your wire. With the holes being that big, I'd be tempted to go with 12 gauge speaker wire from Lowe's or HD for 50 cents a foot, IIRC.
The amp is expecting you to use spade lugs to hook up to its fiendishly hidden connectors. You could solder the lugs to the wires if you wanted to and then hook them up. You could probably find a pair already made up, too, if you didn't want to bother.
Then you run the bare end of the "+" wire over to one red speaker terminal (which is already hooked to the other red terminal, per your diagram) and the "-" wire to the other terminal(s) and you're done!
Thanks for making me take a closer look at my components to make sure I wasn't lying to you. And good luck with your build. I'm still a few weeks off from getting my workshop organized and new sawhorse built, but I can hardly wait!