Do the women of Dorne fight?

Some do. The Sand Snakes, for instance. But it's not the rule.

Nymeria was a war leader but not a warrior -- that is, a commander rather than a combatent.

Or does their "equality" to men only refer to the eldest child being heir instead of the eldest son?

That's the big one, but their customs differ gives women more rights in other ways as well. Not to say that Dorne is an egalitarian society. Far from it.