Hi,
Congrats! Good stuff!
Put considerable effort into your center channel & sub(s). If this is for theater, that's where most of your excitement and fun will come from. Don't fall into the upfiring atmos trap with your budget. You can mount small speakers on your ceiling and aim them at the listening position(s), if you can't do in-ceiling speakers (understandable).
If you're doing Center, L & R, and rears, that's 5 and the 0.1 is the sub (5.1). If you're adding a second pair of surrounds (such as sides, or heights) then that's another 2 channels (7.1). If you ad another sub, that's 0.2, so 5.2 or 7.2 channel. If you add a single pair of atmos, it's another 0.0.2, so your 5.1 becomes 5.1.2 or your 7.1 becomes 7.1.2. In general, only two atmos speakers don't seem to be really worth it compared to 2 pairs, or 4 speakers, to really get the directionality and immersion effect. So that would be 0.0.4 added to what you have. So 5.1.4 or 7.1.4 for example. Things get expensive fast when you try to get into 9 and 11 channel surround on a budget. And all that surround doesn't make an impact the way your center channel & sub(s) do. So keep that in mind.
As for what's necessary, well, that's up to you. Some are fine with a typical 5.1 setup. I would suggest you shoot for 5.2 as a base system and then build on that over time to become 5.2.4 or 7.2.4. But without up-firing versions for the atmos stuff. Take your time on thoughtfully planning atmos if you want a good result.
I would go larger on your sub(s) and if you can double it up, even better. Check out HSU Research subs over SVS.
18x12 is a good space, but its not a large room, so you don't need huge speakers to fill this nor hit listening levels.
The Denon X3700H would be a great source to build from, capable of doing 7.2.4 with an additional amp.
For Klipsch on a budget, I would look at the RP-600C or RP-404C or RP-504C for your center. Again, most of what's going on in a movie, no matter the genre, is coming from this speaker with surround format audio. So unless you're building strictly stereo or 2 channel, put a lot of effort into a good center with a lot of range and capability as this is primarily where the dialog and effects that are not "surround" panned are going to come from. Hugely important. It always gets down played when building a home theater and it doesn't help that all the "sets" of speakers out there always seem to feature a
smaller horizontal center channel which is the opposite of what it should be.
On this note, you don't have to use a center channel, you can use a typical vertical speaker as a center. Explore this option too if you can have a good bookshelf or tower as a vertical center. This can be better than a small horizontal center. Just depends on how your TV/Screen is setup basically if you can have a big center vertical channel.
Save your money and don't get the R625's with the up-firing atmos stuff. Just get a good pair of mains. The mains are important, critical for stereo/2channel, but slightly less as important as the center channel in theater. So put a lot of effort into your left & right. Instead of the R625's. Maybe look at the R-620F's and keep it simple. Again, your space is not huge, you don't need monstrous speakers and towers are more efficient than bookshelves so they will fill your space with sound efficiently. You could look at the RP-6000F's too if you want slightly better.
The R-51M's will handle your surrounds, so you could get 1 or 2 pairs depending on if you're doing a 5 channel or 7 channel setup.
That's the $3k ballpark already.
The rest should go to your subs. Skip atmos for now. Add atmos later after you research your options more.
For the sub, look at a
HSU VTF-3 Mark 5.
Very best,