I sent another email. I hope I'm not coming across as someone desperate for the position, but rather as someone who's really keen on working with them!

If I don't hear back, I'm guessing I should throw in the towel. I have their phone numbers though...would it be out of order if I called them, say, a week from now, if I don't get an email reply? Or would that be creepy?

The optimist in me has been crossing fingers in the hope that my application fell through the cracks, as you put it. I will be totally dejected if I don't hear back (at least an outright rejection will put my mind at ease!), especially as I had believed I had built quite a rapport with my latest interviewer who seemed keen on having me on-site.

This has been precisely my experience. In one case, the hiring manager had indicated interest but HR had not moved to set up the initial on-site interview. After I spoke with her in a phone call, she called them and, violà, interview.

I think it's just fine to follow up. Amongst other things, it shows your interest in working there and with them. You don't want to seem annoying, though. Maybe things have changed, but I'd say a call once a week is about right. And be respectful of their time. Treat it as a status check, and a chance for each of you to briefly clarify any remaining questions. Professional and concise, unless the person on the other end opens it up to be longer and/or more informal. You may leave the impression of someone who does not forget, who follows up, but who does not waste his/her fellow employees' time.

People want to work with people who like and are interested in working with them / on their projects.

Yeah - especially if they are growing quickly and don't have dedicated HR. If it's something on the CEO's / founder's plate then a delay like this woudn't surprise me in the slightest.