Between is a preposition. I is the OBject of the preposition, and should be in the objective case. You should ALWAYS say between my stepmother and me. (The between someone and I is an example of hypercorrection.)

You have other alternatives:

like “animosity between my stepmother and me” which would give you:

“The constant quarrels between my dad and stepmother, along with the animosity between my stepmother and me, created an unpleasant home environment.”

The trouble spot is…envy!

For sentence two, it IS grammatical as you wrote it but “in the sense THAT they” may flow a little better.