What does it mean when partner picks up every single little thing wrong I say or do?

It means they are critical and judgmental.  No thanks, not for me.

Some people are sadistic like that. Whatever you say gets all twisted up.

I was friends with my friends dad and he came up for dinner every friday.

At first we had lovely chats and I liked him like a brother.

Then, he began to be spiteful. He said I must have a screw loose to work at the charity cos I don't get paid.

I stopped seeing him. The invites for dinner stopped. I didn't answer his calls.

His daughter told me he can be very spiteful.

Just see how it goes but if you cannot speak freely about your life it is not a relationship.

It means you are not communicating well. Either you are saying things that are redundant, or the other person feels that they are not being heard. Maybe learn something about communication skills, especially reflective listening. 

“Never love anyone who treats you like you're ordinary.” --Oscar Wilde