How long will a prospective employer wait for you if you're offered an interview/job offer but could not take it right away? I applied to several jobs, but a family emergency came up, and I will be out of town for 1-2 weeks. Should I worry?

If you have already been formally hired, and likely even if you haven't but they would like to hire you, you're probably ok. In my experience, that is the time when you have the most leniency for emergencies and the new employer will have the most patience with you. Also, if you have pre-planned vacations, they will usually cut a deal with you for that time off even if you wouldn't have otherwise been entitled to it. See, they have only seen you at your best so far, so they don't have anything to hold against you. Once you work at a place for awhile, and the "new" wears off, both management and coworkers are less tolerant no matter what the reason. Just like any relationship. Also just like any relationship emergencies are usually completely acceptable, depending on the emergency. The polar opposite would be calling in sick on your first day (which a new employee once did to me.) That's like standing up a first date, there usually isn't another one.

Depends on how quickly they need to fill the position and how qualified you are. If you are a specialist in a field, and it would be difficult to find and interview someone else, they should be able to wait. Also, do you want to work for someone that does not have compassion for family?