In addition to «у меня» being used to denote possession, it is also used to mean "my". So:
собака = the dog ест = eats / is eating у меня = I have / my на кровате = on the bed
I.E. The dog eats on the bed [that] I have = the dog eats on my bed
What I'm not sure on is exactly when «у меня» means "I have" and when it means "my".
The т in "ест" is not palatalized, while the one in "есть" is. The ь (мягкий знак, literally: soft symbol) modifies the consonant in front by moving it further up, i.e. closer to the palate. In effect, this sounds almost like you're adding a sort of subtle "j"-sound. It can be very subtle when your mother tongue doesn't have such a system.
I see that you're learning Norwegian, so a little bonus that you're not going to learn through Duolingo: several Norwegian dialects actually have something similar, particularly some of the dialects in Trøndelag and Østfold. In Trondheim you would for example say "ittj" instead of "ikke", or "mannj" instead of "mann". I've heard that some dialects even have differentiation based on this, where the palatalized form marks something similar to the dative case (which doesn't exist in written Norwegian)
Word order, I think, is very important here. The usual structure to "having something" is у меня есть... and собака ест is clearly a subject-verb construction.
I put both words into the recorded pronunciations by real people at forvo.com and I couldn't hear a difference. The ь really didn't seem to make any difference.
Russian reminds me of French, where there are a number of different spellings of words and phrases, all of which sound alike. There is no difference in the pronunciation of many French verbs in 3rd person singular and plural. il mange = "he eats" sounds exactly the same as ils mangent = "they eat"; Duo gives you credit for using either in dictation exercises. English has any number of the same thing: "fair/fair, pare/pear/pair, hear/here, etc."
There is a distinct difference between a regular т and a ть, though it is very difficult to hear for a non-native speaker (same with ш и щ in my opinion).
Duo probably accepts "est" for both ест и есть since there are many transliteration systems out there - in some systems you would use an apostrophe to indicate a soft sign and double apostrophe or quotation mark to indicate a hard sign; in other translit systems you just omit them altogether from the transliterated versions, though they are still there in the native Russian.
I think that is technically possible but a very unlikely translation. From what I've seen in this course, «у + person» has two uses: to denote possession (with есть/нет), and as an alternative possessor (eg у меня instead of мой). The latter usage without a noun is often shorthand for «у person [дом]».
Distinguishing between these two uses can be difficult but usually comes down to determining whether the main verb of the sentence is есть/нет (or one of these is implied).
However, «у + inanimate noun» is far more likely to indicate location rather than possession.
Of course, I'm not a native speaker, so take this with a grain of salt.
You are right and even more translations fit to "У меня в":
I have a pen in my bag
There's a pen in my bag
A pen is in my bag
All of these can be translated: У меня в сумке ручка.
And still, the common part is "in my bag".
I thought that @tjollans misinterpreted "у меня" as "with me = со мной" and meant to clarify that "у меня" doesn't point to me, eating in bed too, but rather shows that that bed is mine.
as far as I know the propositional case is formed with "-e", so in this case it's an exception?
The pronunciation of кровати here sounds to me more like кроваты or even кровате. In listening to two native-speakers pronounce it at https://forvo.com/word/%D0%BA%D1%80%D0%BE%D0%B2%D0%B0%D1%82%D0%B8/#ru the ending и has a lot more of the long "ee" sound, as in "yankee" or "beet" - just not quite so pronounced.
I get the impression that the computer voice often mispronounces word endings, which are rather important.