"I do not want to invite them to ride horses."
It really should be 不想 not 不要. 要 usually is in response to something that's offered to you (I don't want to eat this, I don't want to go to the park). 不想 is more the intention (I don't want to ask her if she's sad; I don't want to ask them if they think I'm fat)
The word 'them' is not gender specific, so 他 should be accepted. You should not be forced to use 她.
I had the same question on leaving out 去.
"邀" is literary, do you mean "要“？
要 by itself expresses want as in "I want food". So it has a stronger feeling. I think 想要 is closer in meaning to "would like" as in "I would like some apples".