For example, Qantas and British Airways are both One World. If you fly with one, you can usually claim miles with the other (except some award seats).
What about China Eastern Airlines, who code-shares with Qantas, but has their own Eastern Miles program?
The key in Star Alliance is "operated by". For example, at one time SAA was not in *A. Lufthansa operated some flights from Europe to JNB, and code-shared others on SAA. As an Air Canada Aeroplan collector, I would get *A miles if I flew on an LH plane, and not if I flew on an SAA plane. This is kind of moot now that SAA joined *A, but the point still holds for other combinations.
Now, when you're looking at a site like Expedia that shows you a wide variety of flights, they will typically tell you who operates it. When you go to your own airline and ask for a multi-leg thing, for example asking Air Canada for YYZ-FRA-JNB, it can be hard to know who is actually operating the non-Air Canada leg. (Substitute your own airlines and cities as appropriate.) The answer is usually that it is the airline that uses a 3-digit flight number for it. If they tell you the leg is on LH8765 then it is probably not a LH plane; but LH876 probably is. (I say usually because for example Air Canada and United both have some 4-digit flight numbers that are operated by their LCC divisions - Jazz and United Express - but that are treated exactly the same as if they were operated by the mainline carrier.)
Now before you pull your hair out, remember that usually, it doesn't even matter because the operator and all the codeshares are all in the same alliance. It only becomes relevant if you happen to know that some of the codeshare partners on the route are not in the alliance.
To be sure, I will often do a "wide spectrum" search for the flight I'm considering, to let them show me the operating carrier so that I can then check my own plan to see what I will get. It's reasonably difficult to be confident - you have to understand all the fare classes on both your own airline and the operating airline. But you can learn this if you need to and want to.
The answer really depends on the rules of your "native" program, which may allow for accrual whenever flights are booked using the code of a particular partner. Even within the same alliance, however, the rules will vary widely by carrier; in one program you may earn 100% for a certain fare class than only earns 50% in another program.
Whether or not a flight is a codeshare may or may not be relevant depending on the rules of your program. For example, most Star Alliance airlines ignore the code and base earning on the operating carrier. With both alliance and non-alliance partners, the key is to have your frequent flyer number entered into the record (and confirm that it appears on your boarding pass).
You cannot generally earn or redeem on partners of partners of your native program. Qantas is a partner of China Eastern, so you can earn and redeem Qantas points on China Eastern flights. British Airways is an alliance partner of Qantas, but is not a partner of China Eastern, so you will earn nothing. Similarly, Air Canada and Sri Lankan Airlines are codeshare partners, and ANA and Air Canada alliance partners, but you cannot earn ANA miles for a Sri Lankan flight, regardless of how it is coded.
Code share flight have two numbers, one for each airlines. So if XX airlines has a codeshare agreement with ZZ airlines the shared flights will have two number (still, it's one flight) XX123 and ZZ123.
Now if you booked the flight with the same airlines as your loyalty program, you will earn the points just as you usually do. If you booked with the other airlines, ask the reservation agent/agency to include your loyalty program number in the reservation, by doing that the information will be send to your airlines and the points will be earned.
