Help building first tube system.

That's just about the perfect budget for a Raven or Prima Luna integrated. Don't bother with separates. You can't come close to the performance of either of those for anywhere near the same money with separates.

35 or so tube watts is plenty, especially with really smart speakers like Tekton Maob that are also right in your budget's sweet spot.

$3k Swarm subwoofer system, of course.

You would also be smart to skip the DAC/trans and go with CD for the same reason as the integrated is better over separates. Actually you would be smart to go with records but that's another one altogether. Whatever. Point is a solid $3k for source and you still have a healthy roughly $6k for speaker cables, interconnect, power cords, Fuses, ECT, HFT, Total Contact and Omega Mats. 

btw yes I know you said "small 15x15" and "want stand mounted speakers" but I assume that's based on the false assumption that's the way to go in your "small" room. Which is simply not true. Stick the above system in there. When you scrape your jaw up off the floor call and let me know.

I'd second millercarbon's suggestion of a Raven Audio integrated. My Blackhawk LE is the most satisfying piece of equipment I've owned in a lifetime of gear changes. The 45-day in-home, no obligation, free trial makes it an easy call, as well. And the customer service from owner Dave Thomson is wonderful. https://www.ravenaudio.com/product-category/amplifiers/avian-series/

For tubes, Atmasphere- MA-60 with an MP-3 or UV-1.
Vandersteen VLR's
Rega Apollo CD player
Bluesound Node
Ayre Codex DAC.
If you buy used, you should have enough for good AQ cables and maybe a Vandy Sub 3. Though if you splurge on the AQ William Tell speaker cables, you probably won't need a sub.- And, I heard em.
You could save a bit if you went SS and just got a Belles Aria integrated amp. (That's what was connected to the above mentioned speakers/ DAC/Streamer when I heard those William Tell's connected.
Truly a WTF moment.
Bob

How tall are the ceilings? $20 K deserves some travel to a metro area with a few dealers to get an idea on speakers FIRST.

Naim Uniti Core music server $3,000, Audio Mirror DAC $2,500, Manley Jumbo Shrimp preamp $4,600, ATC SCM20ASL Pro mk2 $5,500.  Balance for ICs, PCs, stands, etc.  I believe the Naim doubles as a network player.  Speakers are active, so no amp needed.

This system would be terrifyingly good.  Obviously, everything here is at full retail.  Best of luck building your dream.