Learn French: Le magasin ouvre après dix heures. - The shop opens after ten o'clock.

You could use either, but à quelle heure (at what time) is best here because the answer gives a specific hour (dix heures). Quand (when) is broader and might get an answer like 'today' or 'on Mondays'. You could ask: À quelle heure ouvre le magasin ?

Is ouvre a regular -er verb?

The dictionary form (infinitive) is actually ouvrir (to open), which makes it an -ir verb. However, it is a special verb that copies the endings of regular -er verbs in the present tense. For le magasin (it), we use the ending -e to make ouvre.

Could I just say après dix instead of après dix heures, like we do in English?

No, in French you cannot drop the word heures when talking about the time on a clock. You must always include it, so après dix by itself would sound completely unfinished to a French speaker.

How is dix heures pronounced?

Because heures starts with a silent 'h' and a vowel sound, the 'x' in dix links to it and changes to a 'z' sound. This connecting sound is called a liaison, making the two words run together to sound like 'deez-uhr'.