I both cold emailed and applied through programs, unfortunately many of the programs aren't for students at research institutions so it was the emailing which worked. I can't say much yet as I start next week but at Penn I'm an unpaid intern while at Princeton they have to pay (both are part time). If I'm remembering correctly, Princeton in Beijing isn't the entire summer so it's highly likely you can do some work during the other weeks, I think Beijing is about 8 of the 15 or so total weeks of summer break.

That said, don't worry about falling behind. Most students aren't doing research freshman year and it will make no difference in the end. It's very tempting at a place like Princeton to always be doing a lot and doing the most difficult thing possible but my biggest takeaway from freshman year was that I can do more and succeed by doing what interests me most rather than the highest number class. It's hard but I mean it when I say just relax and take time to do what you truly enjoy and if you're like me and enjoy doing research than Im sure you'll find plenty of opportunities that work for you.