Undecidable problems not Turing-complete? are there systems whose nontrivial properties can't be decided by Turing machines, but for which a Turing machine with an oracle able to find out these properties isn't able to solve the Halting problem (for ordinary TMs)?
Yes, there are intermediate degrees.  One way to see this is that there are computably enumerable sets that are Turing incomparable.  See https://secure.wikimedia.org/wikipedia/en/wiki/Turing_degree for some basic information.