"I want women!"
Haha, yes it is acceptable if you would like to order lady's company from an agency :-D Jokes aside, kérek: I ask, akarok: I want.
And you should keep the accentuated letters as well: kérek: I ask, kerék: wheel, kerek: round.
The translation here of "kérek" as "I want" is not so good. I think it's actually somewhere in between "I order" (like in a restaurant) and "I politely ask for".
Now it is teh only versin they accept - it marks it wrong without the én :-(
No. Please read about the vowel harmony and plural what you can find at the beginning of the lessons here on Duolingo (Tips and Notes).
"Nőköt" follows vowel harmony. The accusative case sometimes takes an -öt ending, does it not? (I believe an example is "tököt".) Perhaps not in the plural? (If so, this should likely be explained in the applicable Tips and Notes. If it already is, I apologize, but I looked and didn't see it.
For the plural forms, the suffixes are always -kat and -ket, never -kot or -köt.