Learn Farsi: من نشست‌وبرخاست با او را دوست ندارم.

This is a coordinate compound noun! It combines the past stem neshast (from neshastan, to sit) and the past stem barkhâst (from barkhâstan, to rise) using the conjunction o (and). Literally meaning "sitting and rising," it creates an abstract noun of action that means "socializing" or "associating."

Why is the object marker râ placed after bâ u (with him) instead of directly after the main noun?

In Farsi, râ points to the entire direct object phrase, not just the head noun. Because the specific thing you do not like is the whole concept of neshast-o-barkhâst bâ u (socializing with him), the râ must wait and attach to the very end of that complete phrase.

Is the verb barkhâstan used a lot in everyday conversation to mean "standing up"?

On its own, barkhâstan is quite formal and literary. In casual spoken Farsi, people normally use boland shodan or pâ shodan to say "to stand up." However, its past stem is permanently frozen into this compound noun, so you will hear neshast-o-barkhâst used even in everyday speech.

I've learned that "liking to do something" uses the subjunctive. Why isn't there a subjunctive verb here?

You only need the subjunctive when dust dâshtan is followed by another verb, as in "I like to sit" (dust dâram benshinam). Here, the action has been turned into a solid noun (). Because you are simply saying "I do not like [this noun]," it functions exactly like saying "I don't like tea" or "I don't like apples," so no subjunctive is triggered.