I know that normally I cannot say "I am having a car", but is the following correct?
Having a car, I decided to drive there.
I am fine with having a car but do not expect me to drive.
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You're right that we can't use Continuous forms of "to have" (= to possess) as predicates (finite forms).
But in your sentences there are nonfinite forms:
Having a car, I decided to drive there. (The Present Participle)
I am fine with having a car but do not expect me to drive. (The Gerund)
So these sentences are grammatically correct.