Learn Italian: Mi dispiace, non posso.

Because dispiacere works exactly like piacere from this lesson! Just as mi piace literally means 'it is pleasing to me', mi dispiace translates to 'it is displeasing to me' (or 'it displeases me').

Could I say io non posso instead of just non posso?

Yes, but it isn't necessary. The verb posso (I can) already has the 'I' built into its ending. You would only add io if you wanted to emphasize that you specifically cannot do something.

Do I always put non directly in front of the verb?

Yes! Making a sentence negative in Italian is very straightforward: just place non right before the verb (in this case, posso). You don't need helper words like 'do' as we have in English.

Can I use non posso on its own like this, without saying what I can't do?

Absolutely. Just like in English, if the other person already knows what you are talking about, you can simply stop at non posso (I can't) without adding another verb to explain the action.