and most other languages in the world, I believe
All the languages you've mentioned are Indo-European, descending from the common ancestor. There are ~450 Indo-European languages out of ~7000 languages in the world, so it's not a good idea to make such a broad generalisation by looking only at one language family.
Well yes. The adjectives-agree rule is not followed in Mandarin and Turkish afaik, nor iirc in Arabic - all not Indo-European. But that's about as far as my linguistic knowledge goes - in fact that's stretching it a bit. I should re-phrase to "most other languages that DL offers" ;-) (Now what about Hungarian?)
You listed three different types of changes that all affect noun/adjective endings in Russian. One is gender, and gender is not going to change--it is an inherent property of the noun (unless we're actually talking about biological gender). The second is singular/plural, and this changes depending on whether you have one of something, or more than one. I believe мои is plural for all genders, but that doesn't mean the gender changes, it just means it's not distinguished on this passive pronoun when it's plural. so that's what's causing the change here. The last thing you mentioned, case, changes when the noun's role in the sentence changes. That's not applicable here. Process of elimination. :-)
Actually, in standard Russian, the pronounciation of «у́тки» and «у́тке» should be absolutely the same (/ʹutkʲɪ/ in IPA, [у́ткь] in Cyrillic transcription), because the second syllable is not stressed and therefore reduced.
So, you can choose the correct variant only based on the context: in «мои́ у́тки» /mɐʹi ʹutkʲɪ/, it's «у́тки»; in «мое́й у́тке» /mɐʹjej ʹutkʲɪ/, it's «у́тке».
Actually, maybe they are distinguished after all. Here's what Wikipedia says:
Across certain word-final inflections, the reductions do not completely apply. For example, after soft or unpaired consonants, unstressed /a/, /e/ and /i/ of a final syllable may be distinguished from each other. For example, жи́тели (listen) [ˈʐɨtʲɪlʲɪ] ('habitants') contrasts with both (о) жи́теле (listen) [(o) ˈʐɨtʲɪlʲɛ] ('[about] a habitant') and жи́теля (listen) [ˈʐɨtʲɪlʲə] ('of a habitant').
So, probably I wasn't correct after all.
However, the wording 'may be distinguished' suggests that they are not always distinguished.
Perhaps we're both correct, and it's possible to pronounce утки/утки with and without the disctinction. However, I don't think the course's audio reflects this distinction.
This one is also interesting:
The merger of unstressed /e/ and /i/ in particular is less universal than that of unstressed /o/ and /a/; for example, speakers near the border with Belarus have the latter but not the former merger, distinguishing between лиса́ ('fox') and леса́ ('forests'), прожива́ть ('to reside') and прожева́ть ('to chew'), etc. The distinction between unstressed /e/ and /i/ is codified in some pronunciation dictionaries (Avanesov (1985:663), Zarva (1993:15))
мои утки. my ducks (plural)
моя утка. my duck (singular feminine)
мой гусь. my goose (singular masculine)
моё животное. my animal (singular neuter)
These are only in the nominative case here, to keep it basic. The genitive case, for example, would be моих, моей, моего, моего, respectively.