I don't think about it often, but I assume Andalites do it like most four legged creature. "Doggy style". As for Hork-Bajir, I assume they do it carefully. It must be possible, and the fact that they have tough hides and heal quickly likely helps.
Now as for Taxxons, they're probably one of those creatures like certain spiders, where the female eats the male. Though I'm not sure if the females are bigger than the males. I'm really not sure how they have sex, or what would keep them from eating their young. Maybe they usually mate near the Living Hive, who obviously can control their hunger. Maybe that's the only place they could actually mate. Or maybe the Living Hive is the only thing that breeds more Taxxons, much like a queen bee or ant.