Can a pilot choose his co-pilot?
As with everything it varies from airline to airline, but for the most part it is option 2). 

Larger carriers will usually give pilots the option of blacklisting another if they wish, but smaller airlines with only a few dozen pilot combinations might just tell the pilot to suck it up. 

The airline will also avoid placing a brand new FO with a brand new captain, aka ‘green on green’ pairings. And on top of this are things like fatigue duty limits, workplace agreements, seniority (bidding), currency and other considerations. Rostering is a very complex exercise.