Do London Underground train drivers all drive on the same line or do they
swap lines?

Just the one line, as a rule (it's not as easy as you might think)

Do you think the drivers all have a favourite line that they want to work
on?

I was on the Piccadilly N'bound at Finsbury Pk the other day, and the
Train Operator announced "this is a Distr..er..Piccadilly line train to
Cockfosters" - wonder if he's moved, or just confused :-)