Is it just me, or this really sounds like 'on ciasto gotuje'
I don't think it does. It may sound a bit too much 'en' than 'ę', but not 'ciasto'.
That's what I thought too, but I still have trouble distinguishing some of the different sounds.
Why is "He frequently cooks" incorrect?
Oversight, added now.
How about "he cooks a lot"?
That's "On dużo gotuje". I understand those can be very similar, but it's still a different adverb.
And he rarely wears trousers.
What about he cooks regularly
Well, "regularly" can be also "He cooks on every last Sunday of the month", which isn't exactly "often" ;)
How about: "he is often cooking"
Words like "often" shouldn't really be used with Present Continuous.