Duo strikes again - introducing an adjective with it's spelling-ending before the module on adjective spelling.
Why Duo does this is beyond me. It causes users to lose a lot of time trying to figure this out, when it would be obvious after studying adjective spelling, which is the very next mudule after "food".
I think in this case it's not too unreasonable. The reason I think this is because 'мороженое' is 'мороженое'; a person doesn't have to know that it comes from the adjective that has the masculine form 'мороженый' to use it as the word for "ice cream". You also don't need to know about declining adjectives for different genders to learn how to spell "мороженое". The only irregularity is in the '-ое' ending, but this is pretty consistently pronounced [əjə] (for those unfamiliar with IPA, something like "uh-yuh"). Just like with learning quirks like '-ся' being pronounced like '-са', you can just learn how the letters are pronounced in combination without need for knowing the grammar surrounding the word.
Other than what's already been said, there is also a phonetic/phonological difference. Russian distinguishes between plain [t] and the palatalised version of [t] in this position, unlike Polish. The plain [t] is like the [t] in Polish "jest"; the palatalised version is pronounced with the middle part of the tongue up against the palate (roof of the mouth). Basically, this is like pronouncing Polish <j> (also IPA [j]) and plain [t] at the same time. The sound is pretty similar to the /t/ in the Polish word <tir> -- just try pronouncing that at the end of the word.
Not in this case. That would be "у неё нет мороженого."
I think the confusion here is related to the fact that the Russian word for "there is/are" (which gets used in possessive constructions and does not conjugate) has the same form as the infinitive of the verb for "to eat" -- they are both "есть." Here, the "to eat" variant is conjugated in the third person singular form ("ест"), and being negated to show that the subject isn't currently or does not generally engage in the activity in question.
The "there is" variant doesn't get negated with "не" to deny possession or existence in the present tense. Rather, it is replaced by "нет," and the object being denied is put in the genitive case.
She could also be vegan or lactose-intolerant.
p.s. she could also be watching her weight, or have teeth that are sensitive to cold; or "she" (Она) could be a duck (Утка) since, in Russian, ducks are grammatically feminine (and should not be eating dairy or refined sugars).
You can clearly hear the double Н in Анна in my experience.
Not to mention that Она is stressed in the last syllable (hence the О that sounds like А), whereas Анна is quite clearly stressed on the first syllable.
I understand it's hard to excercise one's listening skills with a text-to-speech, just try to be mindful of where the stress is everytime.