I work for a company who recently filed for bankruptcy. The store I work at will be closing in the upcoming month. I am a part time employee and I work about 30 hours per week. Can I file for unemployment when the store officially closes while I search for another job?

Depends where you are and how many hours you've been working - the hours question is usually how many hours you've worked over the last year, not how many you had this last week. At 30 a week for a year, you should qualify though, but that also depends on the cutoff in your local area (in the USA, state government does unemployment insurance - in Canada, it's national, but eligibility ranges on where you live.)

as long as you worked there at least 2-3 full calendar quarters, you should qualify for some unemployment benefit - it won;t be a real lot, since you were not full time and IT IS taxable income

Not until you are terminated. There will be quite a waiting period before you actually receive a payment.

You might be better to look for other employment instead of applying.

Yes, you can file - once you're finally terminated.