Doesn't "....'s house" assume an actual house? What if she lives in a flat or even on a boat in a marina?
"At mom's place" would work in both of those cases, and so will Russian "у мамы". In fact, you can drop place/house entirely: "at (my/our) mom's" should suffice.
The difference in pronunciation of мамы by the computer and by native speakers is very definite. Listening to:
it's clear the the ending of мамы sound very close to the way the computer pronounces the subject-pronoun мы = "Mh-ooee", except syllable is very short and cut off at the end. The computer voice sounds more like мы is being pronounced as "mahmeh" instead of "mahmooee" (again, with a very short 2nd syllable). "
According to the following Russian native-speaker with excellent English, y basically means 1 of 3 possible things:
For "near, next to" y is only used with inanimate things, like a school, an office, a home - it's not used to mean "with a person/animal".
"at someone's place". I believe that this is in line with point 1, as the object of the preposition "у" is the (unspoken/unwritten) place, and мамы is genitive because she is the "owner" of that place. "У дома мамы." - "At Mom's house".
"to have", when use with an animate being in the phrase "у [genitive animate noun/pronoun] (есть) [Nominative noun, animate or inanimate}."
"Есть" is in parentheses because sometimes it is not used, but that's another issue.
The video mentions a fourth use for у, but it is quite idiomatic - a usage applying to one verb and one verb alone. I suppose it's mentioned because it may be used a lot in everyday Russian: "Спросить у [Genitive person]" - "to ask something something". There's no logic to this expression, apparently, but there it is.
1) Yours is a question, and it should not be.
2) "Думаю" is in first person, so I THINK, not YOU THINK.
3) "With" is translated as "с" in Russian (i.e. "моя сестра с мамой).
4) You wrote "my mother", but the word "my" does not appear in the Russian sentence, so Duolingo usually won't accept it.