Learn Dutch: Mag ik deze schoenen aantrekken?

You're right that separable verbs usually split up in simple sentences, like in Ik trek deze schoenen aan (I put these shoes on). However, when you use a modal verb like mag (may) in the sentence, the main verb gets pushed to the very end. When a separable verb goes to the end of a clause like this as a full infinitive, it stays glued together as one word: aantrekken.

Can I use aantrekken for putting on other clothes, too?

Yes! Aantrekken literally means "to pull on" and is the standard verb for putting on almost any piece of clothing, including shirts, pants, jackets, and socks. For things you wear on your head or face, like hats or glasses, you usually use opzetten (to put on top) instead.

Why do we use deze here instead of dit?

Deze (these) is used because schoenen (shoes) is plural. In Dutch, all plural nouns act like de-words, so they always use deze for "these" and die for "those". You only use dit (this) and dat (that) for singular het-words.

Why does the sentence start with Mag ik instead of Ik mag?