LR(0) expressive power
You're correct that all regular languages have a LR(0) grammar. Moreover, there are LR(0) grammars which generate a non-regular language. (Exercise: Construct a LR(0) grammar for $\{ a^n b^n\,|\,n&gt;0 \}$.)

LR-ness is a tricky concept to get a grasp on, because LR-ness is a property of the grammar, not necessarily the language. Having said that, it is known that all DCFLs can be recognised by a LR(k) parser in $O(n)$ time. It's also known that any LR(k) grammar can be mechanically transformed into an equivalent LR(1) grammar, and that there are examples for which there is no LR(0) grammar. So LR(1) is the same as DCFL/DPDA, and LR(0) is a proper subset.

So, yes, DCFL/DPDA/LR(1) sits somewhere between 3 and 4 on the Chomsky hierarchy, and LR(0) sits between that and 4. (By the way, don't forget Chomsky type 2 when you're examining hierarchies of languages and machines.)