"He is at her house."
Translation:Er ist bei ihrem Haus.
You have to be careful with the possessive in German. The possessive form for "her" is "ihr" as you have correctly identified. However, you have to further decline ihr based on case. Here, "bei" must be followed by the dative, and the dative form (and neuter, since it comes before "das Haus") of "ihr" is "ihrem."
The reason you're likely confused is because Duolingo does not do a good job of explaining the way you use the genitive of pronouns (think of it as a double genitive: convert the "sie" into genitive "ihr" then convert the "ihr" into genitive masculine "ihrem").
Genitive for non-pronouns is much more straight forward. "The book of the girl" - "Das Buch des Mädchens" (das - des, Mädchen - Mädchens).
Bei vs zu: In this sentence (Er ist bei ihrem Haus) "bei ihrem Haus" does not mean "at her place" but rather "near her house" / "close to her house", referring more to the physical building than to the concept of this house being "her" home.
Zu Hause and possessive: "Zu Hause" is a fixed expression meaning "at home". If you say that somebody is "zu Hause" (or "daheim") this implies that he/she is at his/her own place. If you want to say that somebody is at another person's place, you could say "bei ihm zu Hause" / "bei ihr zu Hause" etc. If you want to further emphasize or clarify that you are talking about your own home, you could similarly use "bei mir zu Hause" / "bei uns zu Hause".
Movement: the equivalent expression for movement is "nach Hause", e.g. "to go home" -- "nach Hause gehen".
"Er ist zu ihrem Haus" does not make sense; if any, it sounds like an unfinished sentence ("Er ist zu ihrem Haus... gegangen.")
No. "am" is a contraction of "an + dem", meaning "at the". Because we have "ihrem" ("her"), we need to say "Er ist an ihrem haus" or "Er ist bei ihrem haus" ("He is at her house"). A different sentence could be "Er ist am haus" or "Er ist beim haus" ("He is at the house"). If you wrote "Er ist am ihrem Haus", it would translate as "He is at the her house", which would be bad grammar.