It sounds like you are looking for small, but it would be much better to buy a used surround receiver off your local Craigslist that has a sub line out with bass management. That way, you can use the built in crossover that has both a low and high pass filter. Otherwise, with the setup you are selecting, you'll have to set the low pass on the subwoofer around the -3db low frequency point for your speakers because a sub doesn't typically have a high pass filter. Plus, if your computer has optical out, you can run digital into the receiver. This will provide better sound unless you have an excellent add on soundcard in your computer. If you are in a city, you should be able to find a surround receiver with bass management for $100 or less.
As for speakers, well, you don't have to limit yourself to Audioengine just because they make multimedia speakers. I use Energy RC-10s
which have a very nice sound. Just be sure to avoid speakers that people describe as fatiguing, harsh, or overly bright if you plan on sitting near them (e.g. be careful of Klipsch bookshelves).
Finally, I don't think the Energy S10.3 will work for what you are currently planning. Without a receiver with a separate sub line out, you'll have to run the speaker out from the Dayton or Audiosource to the sub, and then use the speaker out on the subwoofer to go to the full range speakers. Check the manuals. I'm pretty sure that Polk does have that capability, but not the Energy.
BTW: The Polk PSW505 is $179 shipped (promo code EMCJJJJ36) this weekend only at Newegg. Might want to jump on it while it's on sale. It's a great sub at that price.
Your questions are answered: Speaker FAQ
Living room HT:
Energy RC-50, Ascend Sierra Horizon w/RAAL, Ascend Sierra 1, PSB Imagine XA | PSA V1500 and CHT SS 18.1 | Denon X4200W | modified Dayton SA1000
CBM-170 SE | SVS SB-1000 | Audio-GD NFB-11 | Parasound Zamp | JBL LSR305
Headphone & Portable
AKG K7XX | NAD HP50 | AKG K550 | Soundmagic E50 & E80 | DX50 | E12 | X1