It is correct not to have a "to", but a "to" feels somehow better. If "cleaning" would be used, then the sentence would need to be something like "She helped us with cleaning the house".
Past tense always has male/female different forms. Pomógł for "on" pomogła for "ona"
Possibly because "house" and "home" are not exactly interchangeable, although I think it might be alright in this sentence.
So when the helping verb is perfective, as here, does it matter whether the infinitive is imperfective sprzątać or perfective posprzątać?
Well... "posprzątać" means that the house was clean when we finished, "sprzątać" doesn't mention anything about that - maybe we had such a mess that after a few hours of hard work it is definitely better, but still far from being clean ;)
And "pomogła" just states that on one occasion she helped us with that.