"The cook has a cup of sugar."
Translation:Kokken har en kopp sukker.
It would also be correct to say "Kokken har en kopp med sukker". However to use "av" you would have to say "Kokken har en kopp full av sukker"; which would be like "The cook has a cup filled with sugar".
Yes, for nouns following a masculine declension pattern that is the case: "kokken".
Feminine nouns following the feminine declension pattern get an -a suffix in the indefinite singular ("jenta"), while neuter nouns get an -et ending ("huset").
You can read more about it in the Tips Notes for the skill on definite singulars.