IATA/ICAO airline designations and callsigns

Working in the airline industry you come across a lot of abbreviations and designations for airlines and airports. I've been wondering how some of these designations are decided upon.

Some IATA airline designations are obvious and then there are some head-scratchers.

The obvious ones are AA, UA, DL, KL, LH, AF, AI, AM, etc.

I know not every airline can have fitting designation code, but many seem totally random, like DY/D8 for Norwegian and or NY for Air Iceland. Do the airlines have any say in what their code will or can be, or will they have to accept what IATA decides for them?

I also wonder how callsigns are decided. Again, some are natural, like American, Delta, Lufthansa. But why is British Airways "Speedbird", Chicago Air "Wild Onion" and SalamAir "Mazoon".