Can you keep Bethany after Act 1?

Short answer:

No

Long answer:

If you bring Anders with you, he will inform you of a Grey Warden party who can save her. You'll still lose her as a party member as she has to leave to undertake The Joining. Whether you do this or leave her at home, she will show up in the endgame and potentially fight with you. Regardless, you lose her as a party member. You'll have to get used to Anders, I'm afraid.

Regarding Carver:

He plays out almost exactly the same (doesn't join the Circle, obviously, he has his own twist). Basically, no matter who your sibling is, you lose them as a party member. This is why they don't have specialized talents.

As far as my research into the matter goes, you can't keep her. On top that, she's your sister so you can't romance her.