Learn French: Donne ce livre à ton frère. - Give this book to your brother.

Good eye! Normally, the present tense is tu donnes (you give). However, when giving commands (the imperative mood), regular -er verbs drop that final s for the tu form. So, it becomes just Donne.

Could I say Donnez instead of Donne here?

Yes, but it changes who you are talking to! Donne is the informal tu form, used for a friend, family member, or child. If you were giving a command to a group of people, or someone you need to be formal with, you would use the vous form and say Donnez ce livre à votre frère.

How do I know whether to use ce, cet, or cette for this?

You choose based on the gender of the noun that follows it. Since livre (book) is a masculine word, you use ce. If the word was feminine, like pomme (apple), you would use cette. You only use cet for masculine words that start with a vowel or a silent h, like cet ami (this friend).

If I am telling a female friend to give the book to her brother, would it still be ton frère?