Translation:Continuez à nettoyer.
No, continuer can be followed by à or de. Now the correct prepositions for each verb is something I am still struggling to learn, but in this case, from what I've read, it is most common to use à with the verb continuer. https://www.lalanguefrancaise.com/continuer-a-ou-continuer-de-orthographe Sorry it is in French, couldn't find a good explanation in English.
In a nutshell, "de" used in certain circumstance:
It is the more literary or formal form.
To express something that is a habit rather than an action that is continuing in the present time.
To ease pronunciation. à + a vowel is less pleasing on the ears. Il continue d'apprendre rather than il continue à apprendre.
Here is a helpful list of verbs with their corresponding prepositions: https://www.thoughtco.com/french-verbs-with-prepositions-p2-1364548
Good luck, we'll both need it!
Either should work: continue cleaning/continue to clean, though I would argue that the continuous present is a little more natural to say here.
le nettoyage (the cleaning) is a noun.
Here the English sentence is verb + verb "continue cleaning" as opposed to a verb + noun "continue the cleaning".
Yes, it seems like splitting hairs, but the lessons are designed to teach certain concepts such as having the conjugated verb first followed by the infinitive verb, or when to use the proper preposition (à, de, etc)... The closer you can get to the same meaning within a natural structure, the better.