My mom lives with us. Can we claim her as a dependent?

Your mother is your dependent if she meets all five of the dependency tests explained in What are the dependency tests?.

If she meets all five dependency tests but she files her own tax return anyway, you can still take an exemption for your mother, but she can't claim her own personal exemption on her return.

Even if she doesn't quite qualify as your dependent, you might be able to deduct her medical expenses that you pay for.