First, it's summer. Is her car seat too warm? Are the metal parts hot? You could try draping a light colored towel over the seat when she's not in it.
The way I dealt with this with my grandchildren was to stop trying to buckle them in and just stand there so they couldn't get out of the car. They may have been older. I waited until they decided it was OK to buckle them in. I would tell them, we couldn't go anywhere until they were buckled. Sigh! takes patience and at the beginning 10 or more minutes. Be sure there is nothing in the car for her to play with so she'll get bored.
Another idea is to find a toy that she really likes to play with and only give it to her once she's buckled in and only while she's in the car seat.
The times your husband is not there to read with her, try not reading to her either. Perhaps not even change her diaper. Change it earlier when it's not a part of the routine. When she realizes that crying doesn't get her Daddy there she'll eventually learn that if she wants to be read to she has to calm down. Talk with her about it.
I would let him read to her when he's there and start with her before she has the meltdown. Same with the door. If she has a meltdown you calmly open the door. You want her to learn that meltdowns do not get her what
she wants.
Another way that has worked for me with meltdowns is to, in this example, leave her in the car. Lower a window. Sit down nearby where she can see you and wait until she calms down. The meltdown feels like a temper tantrum.
You could try teaching her to use her voice. Tell her when she asks politely and Daddy is there he will do it.
You could also try taking turns with who reads to her. Talk with her about tonight is Daddy's turn and tomorrow is Mommy's turn. Turns have worked with my grandson who used to be upset when his sister came to my house. Today is sister's turn. Wednesday is your turn. We marked it on the calendar. It did take several months for this to work. Perhaps 2 is too young for her to understand. My grandson was closer to 4 when he accepted turns.