Zur (to the) = zu (to) + dative feminine article (der). Ist das richtig?
zur is a contraction of zu der = to the.
Your sentence talks about some milk in general rather than about "the milk" (that has been mentioned previously).
This is just kinda confusing!
In this sentence Duo's pronunciation of Milch is quite similar to ''mich'', so that's way I heard ''Die Katze läuft zu mich'' at the beginning.
Do seriously natives drop the ''l'' sound of ''Milch'' when pronouncing it at a normal speed in real life spoken German?
No - but the German L is always a "light L"; English would have a "dark L" in that position, so maybe you're not hearing what you're expecting to hear.
Also, zu mich is grammatically impossible since zu requires the dative case.
Because the cat in the German sentence is not merely walking towards (= in the direction of) the milk but is walking all the way to the milk (and ends up next to it).
Die Katze geht auf die Milch zu is one way.
Or Die Katze geht in Richtung der Milch.
Ok, I am just gonna leave this here: If I was to write that "cat goes to the milk'', it would be completely clear to any decent english speaker what I meant. Not to duolingo apparently, so pretty please, diversify Duo's vocabulary concerning the variations of phrases that are usable to solve certain tasks, thank you.
I understand what you mean, but at the same time Duolingo is working towards making you use proper German + English grammar. So if the german sentence includes 'die' for katze, then you want to translate it using the word 'the'. Plus, in English you don't generally say 'cat goes to the milk' you would say 'THE cat goes to the milk'