I just wanted to know if it's possible to switch between the companions. So far I hired Boone and got this little Robot with me but then I meet a girl at a trading post which wanted to tag along.
I'm worried about sending Boone away. Will I be able to re-hire him and if yes, will he stay at my room or will he go back to the dinosaur? What about the other companions I will encounter?

Also does every companion got his personal quest like ED-E or what's his name? Like in Dragon Age / Mass Effect? I would really like to hire all companions at some point (I know I can take only one human along at any time).

Yes, Boone will go to either Novac or Lucky 38 if you have reached it. You can tell him where to go.

Any companion can be re-recruited. However, some companions belong in factions and oppose others: a person named Arcade Gannon won't join you if you are friendly with Legion. The "trade outpost girl", Veronica, will abandon you if you work to destroy Brotherhood of Steel later on. Boone will abandon you if you work to destroy NCR.

And yes everyone gets a companion quest.

If you are on PC there is a mod that allows you to recruit all companions at once without limitations.

Does Lily have a quest besides finding the chewed SB in the Nightstalker cave ?

Never recruited Lily and never will, but her ending mentions about her use of medicine so I think it's related to her stealth boy withdrawal.

She has a quest were you can convince her to take certain amounts of her medicine and its reflected at the end.