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!