About Paris: Please keep an eye on the news before you commit to that trip. The Louvre is being evacuated as we speak ahead of historic floods on the Seine, as well as a number of other museums. The waters haven't hit their crest yet, and no one knows how long the clean-up is going to take once they do. If you're in Paris for other stuff, that might not be an issue, but if you're planning your trip around museums, make sure they'll be open when you go!
I went to Paris this winter for under $350 for four days (airfare included, but I was coming from Europe, so that hardly counts...) and had a BLAST. I was there to see Cathedrals, Museums, and the place where they burnt Jaques De Molay at the stake, and achieved all of those things. I looked at AirBNB and actually didn't find anything compelling. I ended up at a small hotel in the 8th Arrondisment (Hotel Coyopel) -- unfussy, unfancy, but cheap and warm and close to the Metro and a Carrefour Express and a bunch of kebeb places. I'd buy breakfast provisions from the grocery store the night before, eat in the room, and then head out... I got cheap lunch (quiche, soup, etc.) while I was out, and then bought something easy and portable for dinner. I used the Metro and walked (and took a train out to Chartres), and had a completely lovely time.
We did a similar trip to Rome with the kids for a week, with the same ideology and had an absolute BLAST. (We did rent an apartment this time, but we walked everywhere, saw lots of free sites, and lived and died by Carrefour express -- which I find hilarious, because I don't shop there at 'home' because the prices are high.)
If you're looking for super cheap and awesome, I'd totally recommend Eastern Europe. Poland, Romania, Croatia, Solvakia is a huge favorite of all my friends. :)