How far away are we from resolving high temperature superconductivity? What are the major recent findings and their corresponding contributions to an overall picture?
How well explained are the various regions of the dome, is there any thing that is pretty well understood?
What is left and how are people trying to figure it out?
 A: It's difficult to say how close we are to "resolving high temperature superconductivity", as the answer depends very much on your definition of "resolved". For example, have we mapped the phase diagrams? Yes. Do we understand the relevant experimental facts? Yes and no. Is there a complete theory of HTSC that predicts how to create a high temperature superconductors from first principles?  Not yet.  There are a lot of scientists working on these questions, and a lot of very different answers, so it will take some time to come to a consensus, even if the true answer is found today, so you may wish to refine your question.
Nevertheless, the largest "recent" discovery (Jan. 2008) in high temperature superconductivity is the discovery of iron based superconductors.  This is particularly important as iron is nominally ferromagnetic, and magnetism and superconductivity are not natural bedfellows. The Tc of these iron compounds is not close to that of cuprates (highest observed is 55K for SmFeAsO compound, whereas ~150K for Hg-1223 cuprate), but there are many similarities between the cuprates and iron based superconductors which have investigators hoping to find a common mechanism.
There are quite seriously hundreds of thousands of research works on high temperature superconductors, but facts which are pretty well established by experiment are:
1) In cuprates, all static magnetism must be fully suppressed for superconductivity to appear.
2) The symmetry of the superconducting order parameter for cuprates is d-wave.
3) In iron-based superconductors, static magnetism must be reduced for superconductivity to appear.
4) There is a magnetic resonance appearing in unconventional superconductors which scales in energy with the superconducting critical temperature, and this is thought to be a signature of a magnetic pairing mechanism.
And some main outstanding questions remain: 
1) what is the precise mechanism of electron pairing in HTSC.
2) how can we predict superconductivity from first principles - ie, can we create an algorithm which tells us how to mix different elements together to make an HTSC, and what the Tc will be.  
3) is there a common mechanism for high temperature superconductivity, or are there different mechanisms between the different families.
4) is room temperature superconductivity possible?
For technical applications, creating robust wires cheaply and creating materials with higher critical fields is of paramount importance, but that's a topic for someone with more applied knowledge than I.
Nevertheless, as noted in the comments, to understand the open questions in HTSC, one must invest some substantial time investigating the literature.  The most bleeding edge of the research is posted on pre-print servers: ArXiv of Superconductivity Research, while most journals covering physics provide some sort of superconductivity coverage. If your library doesn't have access to those journals, the New Journal of Physics and The Virtual Journal of Superconductivity are open access. Excellent review articles are available in Nature magazine and Reviews of Modern Physics.  
A: Possibly very close.  Peer review needed: http://www.superconductors.org/20C.htm
