Fairly normal; depends on your level of "moderate" use.
No, not really; in fact closing most apps isn't even helpful; only a few (including the web browser) tend to churn unnecessarily, and those ones will usually kill a full battery overnight. I tend to always close the web browser, but the best thing is to install one of the CPU meter applets and make sure nothings hogging CPU.
Like I said before, depends on usage. For me, it was normal (with a stock battery) to charge every night or at most every second night, and often during the day as well, but I use it a lot harder than most.About 6 hours minimum at normal listening volume on my earphones.

If that level of battery life is an issue for you, you might get the 3.6Ah battery from Mugen; it does give over twice the life. Like I said, I use the tablet heavier, so sometimes I wouldn't make it through the day without it, but that's rarely an issue with the Mugen. Not sure it'd be worth it in your case, instead of just getting used to plugging it in more often, since it's already lasting you several days, you can let it charge overnight.