I have Lydia as my companion. She is fully Deadric Armor and weapons. I don't want to lose her as a companion. Level 72. I just killed Alduin. Now I am at Follower to Delphine quest. How do I bring another follower to Delphine without losing Lydia or do I just skip this quest?

Drop Lydia temporarily. She'll wait at Whiterun, iirc. Take whoever else you want to take to Delphine; go back and pick up Lydia. She might unequip the armor and gear, but keep it in her inventory.

Companions quest dropped her before. She was at my house in Whiterun upstairs. I will take her back to my house in Whiterun. Can I use wait here ? What is the best way to drop her?

I don't believe you can use wait. That would mean she is still in your service as a follower, and you won't be able to get another.

Dismiss her; take her to Whiterun first, if you want, then tell you its time you part company. She's your Whiterun house carl, she'll stay in your house in Whiterun until you get back.

Thanks. I will try it.