I would like to pick out small objects from a category.

Question 1. Are the compact objects of the category of $k$-schemes exactly the schemes of finite type over $k$? If not, what are the compact objects?

Question 2. What are the compact objects of the category of derived schemes?

Question 3. What are the "compact objects" of the category of topological spaces (equipped with the Quillen model structure)?

Question 4. What other notions exist for "small" objects?

References or comments are welcomed.:) Thanks!

I would like to pick out small objects from a category. I would like to find such a notion which

Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least picks out something relevant.

Dream 2. Picks out the top spaces homotopic to finite CW-complexes from the category of topological spaces (equipped with the Quillen model structure). (or something relevant)

Question: What notions exist for "small" objects (other than the compact objects)?

References or comments are welcomed.:) Thanks!