First of all, congrats on soon getting the SongTower II speakers. I got SongTowers in 2007, and replaced them in 2016 with Veracity ST speakers. I love them, and I'm fairly certain you'll love yours too.
I generally agree with what
@ski2xblack says above. To address each of your questions.
1. AVA Set 120 Control Amplifier ($1300)
https://avahifi.com/collections/power-amplifiers/products/set-120-control-amplifier
Is this a good choice? Any other suggestions?
That integrated amp is a very good one. It will be plenty powerful for you SongTower IIs. You won't go wrong with it. However, you can find other integrated 2-channel amps that will also be fine for your speakers, at a lower price. I agree with what
@ski2xblack suggested: Outlaw or Yamaha integrated amps or receivers. Any amp that delivers at least an honest 50 watt/channel should do well. Fortunately, there are plenty of such amps in the 50-100 wpc range.
Many of your questions (below) lead me to suggest an audio/video receiver (AVR) instead of a 2-channel integrated amp. Think carefully about what kind of inputs you will need. The AVA SET-120 has 4 pairs of analog RCA jacks. It has no DAC, no built-in phono pre-amp, and no Wifi streaming, and no remote control. Will that be enough for you now as well as in the future? It sounds to me that you aren't sure about what you may want in the future. Once you pay for that AVA integrated, you may not want to replace it soon, so consider things carefully.
2. Google Chromecast Audio ($50)
Will I be losing much quality with this? Should I get an amp with a DAC built in?
No problem with Chromecast Audio, but if you want to do that, you should think about an integrated amp with a built-in DAC. Without a DAC, you couldn't connect it to the AVA integrated.
3. HiFi streaming service (at least until Spotify introduces its own)
How would you connect that to the AVA integrated?
Any decently constructed and decently priced RCA interconnect and speaker cables will be just fine. Look at Monoprice, Blue Jeans cable, Amazon, etc. There is no sound quality benefit if you pay more for exotic cables.
Later on, what do you suggest considering? Some ideas:
1. A center speaker: Maybe not that useful if I'm so rarely watching movies?
For that, you will do better with an AVR. Same answer for 2 surround speakers. No stereo integrated can handle them as well as an AVR. I don't see how you could connect TV or movie sound to that AVA integrated.
However, before buying a center or surround speakers, you should first listen to how well your two Songs do by themselves.
2. A subwoofer: I'll likely be living in an apartment building with units underneath, so perhaps not that necessary and not that neighbor-friendly?
Before you consider a subwoofer, listen to your SongTower IIs in your home. For music, those speakers do bass well enough without a sub.
4. Phono preamp or 5. Turntable: I currently have no records of my own (though parents do), so seems like something that can wait, unless this could be included in some amps?
If you don't already have records of your own, I'd say, don't bother with buying them or buying a turntable, pick-up cartridge, or phono preamp. It's not worth the cost or the fuss.
I suggest you look into an AVR with about 100 watts/channel (when 2 channels are driven), that includes a DAC as well as several HDMI inputs and outputs. Look for AVRs made by Denon, Marantz, or Yamaha. They will allow you to digitally hook up TVs, disc players, streaming devices, etc., and become the switching center for all those audio or video sources. And it will have plenty of power to drive your speakers.
If you have two speakers and do not plan on a center or surround speakers, consider the
Denon DRA-800H. Its a 2-channel receiver with five HDMI inputs, that allow you to hook up the latest TV/movie audio & video, wireless streaming, etc. Its basically a full-functioned AVR with only two audio channels instead of 5 or more. It produces 100 wpc and costs $700.