Learn Portuguese (Portugal): Fazemos a patrulha para afastar qualquer intruso. - We do the patrol to keep any intruder away.

Because the subject of the main verb (Fazemos) and the purpose clause is the same ('we'). In Portuguese, when the subject doesn't change between the main clause and a prepositional clause, it is standard to use the basic, impersonal infinitive.

Why is para used here rather than por to mean 'to'?

Here, para introduces a purpose or goal, meaning 'in order to'. Por would indicate a cause or reason ('because of'), which doesn't fit the context of doing a patrol to achieve a specific outcome.

How does the word qualquer behave if we change the sentence to refer to multiple intruders?

It would become quaisquer intrusos. The word qualquer (any) is invariable for masculine and feminine, but it has a highly irregular plural form: quaisquer. However, keeping it singular (qualquer intruso) is the most natural way to express 'any intruders' in this context.

Could I use evitar instead of afastar in this sentence?

It would slightly change the meaning. Evitar means 'to avoid' or 'to prevent' an event. Afastar literally means 'to push away' or 'to distance', implying the patrol will actively repel any intruder that tries to approach.