What is the difference between hejmo and domo? It switches half way through.
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..
Dankon, David! I realized that I'd written "home" rather than "house" the first time around, and hence misattributed the error. Your response helped to clear this up. I've since tried it again and this sentence is indeed accepted. Hooray!