от requires Genitive, as do many other prepositions (e.g., без, с, у, возле, около, для, из-за, вне, мимо).
The Prepositional is ONLY used with prepositions, but most prepositions are not used with this case. In fact, it is only used with в, на, о, при and, in some set phrases, with по (which is normally a Dative preposition).
Usually, из is used for a place. For example, Я из Санкт-Петербург. It would be incorrect to say Я от Санкт-Петербург. In general, when the word after 'from' would be a place, you should use из and if it's from an object or from a person, we use от. To quote it best, от is from a specific point. из is 'coming out of something'. I am from St. Petersburg = I come out of St. Petersburg. You do not 'come out of a door'. If you can say 'I come out of Insert word', then you should probably use из.
I'm still struggling with pronunciation of cyrilic letters, so it might just be that I don't know enough yet to form a sensible question, but here goes...
The pronunciation of двери completely surprised me here. I would have expected something like "dvyeree" (pardon my phonetic spelling, not sure if that makes sense to anyone). Instead it sounded like "dneega" to my ears.
Is this just an oddity of this particular word, or a mangled TTS sound?
I do hear "dvyeree", at least something much closer to that than to that "dneega" thing.
Something that I hear is that the "ть" ending of the word "кровать" seems to be pronounced here clearly as a "т" (hard consonant) no matter how many times I listen to it, while I usually can hear the difference between т and ть (at least after repeating the audio if necessary). Is the audio fine here?
Most English speakers have problems hearing the difference between hard and soft Russian consonants because we don't hear them while we are growing up. Up to about 8-10 months, children supposedly can hear and differentiate any kind of speech sound. After that, the sounds that may vary in "correct" pronunciation but represent the same phoneme are replaced by something that is sort of in between, but in any case are treated the same and are not discriminated between.
However, in Russian, the soft and hard versions of a consonant are significant and words which differ in them mean different things. The most notorious example is "мать" and "мат". "мать" means "mother" while "мат".means "filthy language". ;-)
Pronunciation is a skill learned not only with your mouth, but also with your ears. It is important that you know the pronunciation of all the Cyrillic alphabet VERY well, so well that you can easily read any word (even big words you don't know the meaning of) at a good pace. At this point, your brain is trained well enough to handle exceptions with ease.
I am learning Russian as well, & I barely have any issues with pronunciation & hearing the words. I now purposely practice with low volume or in crowded areas to make it more challenging to hear the words.
But, how do you perfect pronunciation & hearing? Easy, find any Russian text or typing course online. You can either read or type these exercises (typing increases yet another skill!)... For each word, read out-loud the sound of each letter to the beat of a metronome (free metronome when searched on Google). Learn to say the names of every character & read those sometimes, too (example: “точка" = period. "пробел" = space. "тире" = dash. "равно" = equals. "запятая" = comma. "Мягкий знак" = soft sign [ ь ]. "твёрдый знак" = hard sign [ ъ ].) Try to read a full page this way every day. After 1-2 weeks of doing this every day (or whenever you are comfortably reading out loud each character/letter), now try reading each syllable out loud.
If you are unsure of your pronunciation, join "Hello Talk" app and ask people to pronounce or correct your pronunciation each time. Or have a web page with audio recordings open to assist you when you're stuck. Eventually you'll be able to read everything comfortably on your own. When all the basic letters are done comfortable, exceptions are easier for the brain to pick-up, because they are only a little extra weight added to a light load rather than added to a heavy load.