amp (or receiver) recommendation for Polk Audio RTi A7 floorstanding speakers

For a power amp, I'd suggest for consideration a nice used McCormack DNA-1 Deluxe if you can find one in great condition.  Sometimes they can be had for ~$700.   150 watts per channel. 

A great and versatile pre-amp(which I'm very interested in) is the Parasound Halo P5, which does pretty much everything and has home theater bypass.  They run about $1000 new, but I've seen some here on Agon for $700 and up.

To me, a "basic" system would consist of my Polk RTi A7’s and a good integrated amplifier (or preamp + amp). Is that reasonable?

An integrated amp may be the best way to go since it takes experience to match separates such as a preamp and power amp. An integrated with a built-in DAC would make sense; an all in one solution. I’m not up on the latest amps, but you will need adequate power to drive your Polks or whichever speakers you choose.
What will your source be...CDP, streaming, computer files?

OK, you seem to have reached a satisfactory solution for the small room and HT, using, as I uderstand it, the two smallest Polk speakers.
That leaves the music room question. The thing you and only you can decide is in which room you will wantto listen to music. Both are suitable, but the bigger of the two potential music rooms will allow even deeper bass to be reproduced without too much degradation. But in essence this is a lifestyle issue.
I can understand that you do not (yet) want to splash out on expensive speakers (I did not quite realize the Harbeth speakes are so expensive in the US, and I did not know exactly how much you could get back if you returned the bigger Polk speakers).
So, I assume you will want to keep one of the two pairs of Polk floor standing speaker pairs. In the bigger room I would opt for the biggest of them, in the mid size room I would try what you prefer.
And those speakers will need an amplifier. My advice would be to buy something that is not too expensive, but good enough to be used completely succesfully when you decide to upgrade the speakers. Fortunately, and contrary to what some here believe, good electronics do not need to be expensive.
For the mid size room I would recommend the Yamaha AS 801. It performs flawlessly (see the test results of their AS 700 that I posted earlier - essentially the same amplifier but wthout the modern digital inputs) and 2x100 watt will be enough to drive almost any good speaker in this mid size room. Harbeth’s Alan Shaw bought the almost identical AS701 for a demo room at the forthcoming Bristol show. Having seen the test results he did not even open the box and will not do so until the day of the show. So much for burning in or long listening tests to establish ’synergy’. He will not use fancy cables either. He is one of this world’s most respected speaker designers.
For the large room something more powerful would be required once you move up to more ambitious speakers (with your current quite sensitive Polks the AS 801 may still be just OK). But a room like this does indeed require big power, and that does not often come in integrateds. Hence my suggestion for something like that 2x350 watt Yamaha pro audio amplifier with either the Pioneer U-05 or the Oppo 205 as a front end, depending on whether you want to play discs or not. Both have balanced XLR connections for lower noise. One of these big powered combinations should be more than good enough to drive any high quality speaker (up to Harbeth’s top of the line M40.2) in a big room. I bought the 2x250 watt P2500S for my son’s birthday as he is planning to buy a pair of Harbeth M30.1s by next summer, and the resulting sound with the current speakers is as clean and precise/neutral as you could wish for. If you want to be absolutely sure, you can even go one size up from the P3500S, to the 2x500 watt P5000S. Harbeth recently demonstrated their big speaker in the Netherlands, and the power meter on the amplifier that they used indicated that at times it produced more than 500 watt on peaks of dynamic music. I just saw a P5000S on Amazon for $650. These excellent pro audio amplifiers can be so cheap because unlike audiophile stuff they are produced in huge numbers. Audiophiles do not want to know, and the cottage industry works hard to persuade everyone that somehow pro audio is no good, but proper tests show otherwise. Use these big amps with your Polks for now and you do have to be a bit careful, but pro audio amps have their own gain control and on the Yamaha’s you can set that behind a screw-on lid to prevent visitors (like my son’s friends) from destroying your gear. Similarly, they have adjustable high pass filters to keep the very lowest frequencies from wrecking your speakers.

I have been following both your threads and you have been given much great advice. I agree with the path you have taken towards a solution by separating the video and audio rooms and using a single pair of speakers for audio.

Your sources are a disk spinner and your computer. Lots of ways to improve on these, however as you are considering a basic setup for now, and one that you will use to 'tune the room' via treatments, you need not concern your self with that now. 

The next step is to purchase amplification for your speakers. With your budget I would encourage you to purchase a integrated amplifier. There are many within your budget that are quite good. Based on what others have said in previous posts it seems that your speakers are demanding of power. The previously mentioned Yamaha 801s delivers 100wpc into 8ohms and 200wpc into 4 ohms. It has onboard digital (PCM and DSD) via USB, optical, and coax, and be found new under your $1000 budget. It is not the only choice, however one that I am familiar with (not owned) and may meet your needs.   

"What will your source be...CDP, streaming, computer files?"

Some kind of CD player and computer files played directly from the computer (via USB or HDMI or optical digital audio out on the computer), or I could pass the computer's audio out through a Focusrite Scarlett 6i6.