Learn French: Je vais le rattraper dans dix minutes ! - I am going to catch him in ten minutes!

This is the core difference between these two prepositions! Use dans to talk about when an action will happen in the future (the waiting time until it occurs). So here, you will catch him 10 minutes from now. You use en to express how long it takes to actually complete an action (e.g., Je l'ai rattrapé en dix minutes - I caught up to him in a span of ten minutes).

Why is the pronoun le placed between vais and rattraper?

When you have a conjugated verb followed by an infinitive (like the "futur proche" here with aller + infinitive), object pronouns always go immediately before the infinitive they belong to. You are catching him, so le goes right before rattraper.

What is the difference between attraper and rattraper?

attraper simply means "to catch" (like a ball, or a cold). Adding the prefix r- to make rattraper usually means to catch again or to catch up with someone or something that is getting away, which fits perfectly here.

Could I use the simple future tense (Je le rattraperai) instead of Je vais le rattraper?

Grammatically yes, but the "futur proche" (vais + infinitive) is the most natural choice here. It is frequently used for events that are going to happen in the very near future—like in exactly ten minutes!—and for actions you are highly certain about.