Or maybe... her brother was just thinking that she could take his car and then he'd visit with someone to see that she has settled in nicely and would then take his car back home.
She wants to continue on her "independence" by not needing her brother's help. Good for her. You can see that her choice to live a healthy life of independence includes her immediate family and not just you. And do understand that type of independence does not exclude a M.
It certainly was nice of her to offer to take care of getting your car checked out.
As far as planning things with her, she did ask about the friends and maybe she'd be interested in going to the beach, so you could eventually let her know that she was welcome to join or go with.
Otherwise, just plan YOUR life as (now) "normal" and put invitations out there now and then for some of the things you think she might enjoy. Just don't plan ONLY things that SHE would enjoy. Plan things FOR YOU.
