That's kind of routine stuff. Ree can explain it better, but it is similar to a mountain bike. It was loose before, her pantleg complicated it. There are a couple of small 'levers' on the back wheel. You push (rotate) these till your chain is good. A lot like a 10-speed bike, maybe a little easier. Make sure your chain is oiled (dry chain is bad!). That could throw it too.
Like they said, wow, 4 or 5 links is a LOT. I've got a Tomas too, But I dont think I could keep the chain the sprocket on if it were that loose!
You'll get a flat sooner or later-- Your only good luck seems to be having a girl on the back of your ped!
Tightening the chain is what you do after you change the tire, so it's good to learn.
I was thinking of using one of those 'slime' tire additives, supposed to seal punctures.
good luck- Keep us posted