Learn French: Pourquoi veux-tu que nous ayons un règlement strict ?

The phrase veux-tu que expresses a wish or desire, which is a classic trigger for the subjunctive mood in French. Because you are wanting someone else to do something, you must follow que with the subjunctive form of the verb, which is ayons instead of the indicative avons.

How irregular is avoir in the subjunctive?

Highly irregular! While most verbs form their nous and vous subjunctive by looking exactly like the imperfect tense, avoir uses a completely different stem. Its forms must be memorized: que j'aie, que tu aies, qu'il ait, que nous ayons, que vous ayez, qu'ils aient.

Could I avoid the subjunctive and just use the infinitive, like Pourquoi veux-tu nous avoir...?

No, because the two verbs have different subjects. You (tu) are the one wanting, but we (nous) are the ones having the rules. When the subject changes after a verb of desire like vouloir, you are forced to use que plus the subjunctive. If the subject stayed the same, you would use the infinitive: Pourquoi veux-tu avoir un règlement ? (Why do YOU want to have rules?).

The English translation says "strict rules", but the French uses the singular un règlement. Why?