I would start with updated appliances and possibly a focus over stove. The counter tops and splash could be upgraded to a more current material and look.if you are going for a contemporary look glass tiles would look great. It would be helpful to have a plan so you could execute a cohesive look and do it in stages. I would suggest talking to a professional kitchen designer even if it is at a big box store.

I think it would make both rooms too dark. Maybe go with a lighter floor colour?

You should remove the cork first as tiles need a flat and even surface to go on. Definitely remove the cork.

I think like you -- I wouldn't spend $10k to make new cabinets just to centre things . A couple of 'thinking out loud' thoughts -- is it worth adding another layer of wallboard -- I'm thinking ply rather than gib -- then tiling over it ? It may stiffen everything , and ply has less flex , and as you are tiling over it , the finish doesn't matter too much , and the fact it would add say 18mm to the wall depth would be hidden by the fact that its to be tiled anyway . You may have to add a 'wider' facing around the doorway so it hides the extra thickness , but relatively easy to do ? Or is it worth looking at changing to a solid door ? Yes , I know it cuts space a bit , but depending whats on , say , the left hand wall , it may work , and its another way of stiffening up the wall , although it would need a builder .