Learn French: Où est-ce que vous mettez la souris ?

You spotted the pattern! The verb mettre (to put) is irregular. In the singular forms (je mets, tu mets, il/elle met), it drops one 't'. But in the plural forms, it keeps the double 't' from the infinitive: nous mettons, vous mettez, ils/elles mettent.

Can I just ask this by saying Où mettez-vous la souris ? instead?

Yes, you can. Où mettez-vous la souris ? uses a structure called inversion (swapping the verb and subject pronoun), which sounds a bit more formal. Adding est-ce que after the question word Où is the standard, most common way to ask a question in everyday spoken French.

Does la souris mean a computer mouse or a live mouse here?

It can mean either, just like in English! French uses the exact same word, la souris, for both the little animal and the computer accessory. You have to rely on context to know which one someone is talking about.

Why is there an 's' at the end of souris if it is just one single mouse?