Andremo in gelateria solo quando avremo ripulito la zona attorno all’ombrellone.

solo quando means “only when.” It adds an exclusive nuance, showing that the main action happens only after the subordinate action is completed. By contrast, plain quando simply means “when,” without that exclusivity.
Example without exclusivity:
Andremo in gelateria quando avremo ripulito… (We will go to the ice-cream shop when we have cleaned…)
With exclusivity:
Andremo in gelateria solo quando avremo ripulito… (We will go to the ice-cream shop only when we have cleaned…)

Using andremo (future) clearly states a plan or promise about what will happen. In casual spoken Italian you might hear andiamo used for a future outing (“let’s go”), but in standard Italian the simple future emphasizes intention or scheduled events:
Andremo in gelateria = “We will go to the ice-cream shop.”

In Italian, when the main clause is in the future tense, a temporal clause introduced by quando often uses the future perfect to show that one action must be finished before another future action. Here, the cleaning must be completed before going to the gelateria. Hence:
avremo (future of avere) + ripulito (past participle) = future perfect.

The future perfect combines:
• The future of the auxiliary verb (avere or essere)
• The past participle of the main verb
In your sentence:
avere (future) → avremo
past participle of ripulire → ripulito
→ avremo ripulito

In Italian, when talking about going to a type of shop, you often use in + shop name without the article:
andare in gelateria, in pizzeria, in farmacia.
You could use alla gelateria if you mean a specific shop (“to the gelateria on Main Street”), but in gelateria is the usual general expression.

Yes. You can front the time clause for emphasis:
Solo quando avremo ripulito la zona attorno all’ombrellone, andremo in gelateria.
Just add a comma after the subordinate clause; the tenses and words remain the same.