Learn Portuguese (Portugal): Abre os olhos devagar.

Abre is the informal command for tu (you). To make a positive command for tu, you take the present tense form (tu abres) and drop the final 's'. If you were speaking formally to someone using você, you would use abra.

The English translation says 'your eyes', but the Portuguese uses os (the). Why is there no word for 'your'?

In Portuguese, when you talk about body parts, you usually use the definite article (o, a, os, as) instead of a possessive word like teus (your). Because you are telling the person to open the eyes, it is already obvious whose eyes they are!

I thought adverbs like 'slowly' usually end in '-mente'. Why does devagar look different?

Devagar is a common, irregular adverb meaning 'slowly' or 'slow'. While you could use a regular adverb like lentamente, devagar is very common in everyday speech and does not need a -mente ending.

Can I add the word tu and say Tu abre os olhos?

No, when giving a command, you do not use the subject pronoun. You just start directly with the verb. If you added tu, it would sound like a statement, and you would need the present tense form instead: Tu abres os olhos (You open your eyes).