Grammatical issues help please?

Hi there,

Can you confirm me that I am correct in these grammatical issues, please?

A lot of bother finding it. (Bother is a proposition)

I have a hard time finding it. (Hard time is a proposition)

I had no luck finding it. (No luck is a proposition)

I had no difficulty/trouble/problem reading it. (Difficulty/trouble/problem are all the proposition)

I don't mind doing that. (Mind is a verb)

That car goes fast. (Car is a noun therefore it doesn't follow with ing)

Cheers.