Can Kakashi use the jutsu he copied without the Sharingan? I'm not talking about Sharingan specific moves like Amaterasu, but the ones he had copied with the Sharingan like Water Dragon Jutsu.  

With the sharingan he has copied and can use over a thousand moves.  Lets say he no longer has it (removed), does he retain the skills he had copied?

Does it depend on Sharingan or the skill of the user?
It's a good question you there. Kakashi uses the Sharingan to copy ninjutsu that he has seen from other ninjas like the Water Dragon Jutsu and to continue using them, he has used his memory and experience to remember the seals to perform them. The Sharingan is a tool which he uses it to learn other ninja's ninjutsu the fast way and his brain memorises the ninjutsu so that he can cast it instantly whenever he needs it.

Hope I have answered your question.