In myths or legends, are there certain demons that rule certain parts of hell?

There are way too many to name. If you look at Goetic demons( demons that were bound by king solomon and demons that most demonologists research) you'll notice that most demons listed have some sort of title (King, Duke etc.) and all rule over different bits of hell as well as many lesser demons

Here's a list of some of them: http://en.wikipedia.org/wiki/Category:Goetic_demon...

If you want more information it shouldn't be too hard to find by looking up the lesser key of solomon or geotic demons via google.

According to The Lesser Key of Solomon there are 4 crowned princes of hell

Lucifer

Beelzebub

Azazel

& Astaroth

I've also seen stuff about "The Seven Princes of Hell" where they have demons that are supposed to represent the seven deadly sins. (They're supposed to be very powerful)

Pride --- Lucifer

Greed --- Mammon

Lust --- Asmodeus

Envy --- Leviathan

Gluttony --- Beelzebub

Wrath --- Amon aka. Satan

Sloth --- Belphegor

You'll find different myths about demons in all cultures if you're curious you can just look up different cultural and religious beliefs and you'll probably find something.

No. Satan rules Hell. The demons are other fallen angels.