The accusative is typically reserved for the direct object of a transitive verb. The direct object is never the object of a preposition.
Mi pentras la domon. = I am painting the house.
What is being painted? The house.
Mi iras al lia domo. = I am going to the house.
What is being gone? Doesn't work. "Lia domo" is not the recipient of any action, it is merely the destination. If this were Latin, it would take the locative.
The only time the accusative is used after a preposition in Esperanto is when it is necessary to disambiguate the kind of motion going on.
I jump on the table. = Mi saltas sur la tablo.
I stand on the table and jump up and down.
I jump onto the table = Mi saltas sur la tablon.
I stand elsewhere, jump, and land on the table.
Not necessarily. According to PIV (Plena Ilustrita Vortaro), one meaning for "marŝi" is "Paŝadi iom longan vojon, por promeno aŭ alicele" (That is, "To walk some long way, for a walk or with another aim". But it wouldn't be correct here, because the verb is "going", so it is more general, and doesn't specify whether it is walking, crawling, swimming, flying, skiing, or whatever, so in Esperanto that would be "iras".
A house is a specific type of building. A home is anywhere you might live, including apartments.
A house is the physical building. A home is what that building (or part of a building, or tent, or igloo, or whatever) becomes when someone lives in it.
Yes, that should be acceptable. After all, if they were statements, not questions, "You go to his house" and "You are going to his house" would both be "Vi iras al lia domo" in Esperanto..