If I am booking a flight for somebody else, is it usually possible to get frequent flyer credit for that flight?
Using American's booking interface, it complained when I tried to use my FF number with the other passenger's name. So, is the policy that only the person who flies gets the credit, not the person paying?
Generally, frequent flyer programs give credit to the passenger who actually flies. The person who books or pays for the flight, if different from the passenger, doesn't get anything.
A major reason for the existence of such programs is as a "kickback" to business travelers. The company pays for the ticket, but the traveler gets the credit. This is supposed to encourage the traveler to prefer that airline, even when another airline might be cheaper (and the traveler doesn't really care about the price because the company is paying).
As a rule, no, you don't earn miles for flights taken by others, even if you're paying the bill.
However, a number of airline programs do have ways to accomplish this in practice:
A few airlines, notably Etihad and Virgin Australia, have a "Family Membership" program where the nominated Family Head gets to pool all miles collected by family members; see this question for a partial list. I'm not aware of any American airlines supporting this though.
More common is the ability to transfer miles from one account to another, so your "somebody else" can collect the miles and then forward them to you. For example, I use this ability on Qantas to hoover up my kids' miles, since they can't make use of them anyway. Often this is only permitted between "family members" or even people living at the same address, but in practice there are no checks on this, other than the computers getting suspicious if you're accumulating new cousins on a weekly basis (this feature is sometimes abused as a way to sell miles to third parties, which is prohibited). American has this option too, although it's a paid service and the costs are kind of prohibitive at $12.50 per 1000 miles.
The person who flew the flight you paid for can also credit back the miles by purchasing something for you with those miles, whether it's a flight, a gift card or some overpriced knick-knack from the rewards catalog.
The main catch with all of these is that you can't earn status via somebody else's flying: to reach (say) a 50,000 mile Gold status, you have to fly those miles personally.
I know of no program where the credit goes to anyone but the traveller, although you can debit miles for a ticket to pretty much any another person (i.e. use miles or points to secure a ticket for someone else). In some programs you can transfer miles between program members in the same family, but that's about it.
Indeed airlines rely on the total number of miles being fragmented over a very large number of travellers so as to reduce the number of people with sufficiently many miles in their individual accounts to redeem a reward ticket.