Delta is useless, forget them. However, AA is a good program to collect miles in.

AA has something called All-Airline awards, which are point-to-point and you can use their partner airlines as well. (Jet is a partner in India, Kingfisher will be when it joins One World.) You need to call AA and ask for an award from LAX to BLR. They'll figure it out. Besides AA (or any partner) to DEL and Jet to BLR, you can also connect via London. IOW, there you won't need to buy an extra DEL-BLR ticket.

However, the London route will have high taxes---and thus better availability ---and should be reserved as the last option.

Re CX bonus, each airline has its own programs. The CX bonus will not be available on AA. If you can, you should definitely fly CX to HKG. It is a HKG based airline and one of the very best in the world. You can't possibly do any better.

PS. Welcome to FT!