That is very much correct.
And for this rhyme you've my respect!
很 is not only "very". It is used to connect subject to an adjective.
Is the 的 really needed here?
That was my question too. I am under the impression that with singular family members (including pets) the 的 isn't necessary. Anyone else want to comment?
From other comment sections it seemed that pets aren't considered close enough to drop the 的, either that or 我狗 and 我猫 sound weird.
No, it's not. For almost all other questions it accepts it without the 的. It's a glitch!
Oh lawd he comin
Rude but ok
There was no "very"?
很 is just used to complete two syllables, doesn't really mean "very". If you wanted to say "very", you would use 非常
The 很 is used for grammatical reasons here, so it doesn't really mean very.
Alternatively you could say 我们的猫非常胖 to express that the cat is really unusually fat.
很 can often be used in the same way as "is" would in an English sentence.