The pattern of the layout of the tile straight vs diagonal.  

Fortunately they do and then some.Solar reached "grid parity" across the USA generally about three years ago with fossil fuel power.

Once the refi process is finalized should I expect there to be parity between the improved value figure that the lender assigns and the improved value figure that my county auditor assigns? 

Chris with your first picture, the 1 x 6 on the diagonal is your sub-floor, then the oak is on top of that. you can just screw the cement board to the oak floor but the thinset as i said helps to support the 1/4" cement board because it is thin and will follow any bumps in the floor, the thinset helps to level it off a little.

I am also trying to figure out if I can fit a dishwasher diagonally in the corner next to the stove.

And diagonal cracks on blocks (like a stair case) coming from the bottom of the pier?  

Mortgage rates are still very low, unemployment is below parity in our city, and vacancy rates are still very low.

This is happening in many markets nationally right now.Right, and many of those markets are vastly overvalued and are as likely to correct as anything else that is vastly overvalued.Your long term diagonal trend line is overoptimistic and unsupported by historical precedent.Real estate does not appreciate over the long long term, nor would logic dictate that it should.I'll grant that certain areas do improve and therefore show an increase in intrinsic value but 1. this should be supported by incomes and 2. predicting this type of change is extremely difficult.

Look for recently repaired brick mortar, especially diagonally from the corners of the openings.

From a legal liability standpoint, should I be concerned about rent parity among tenants in the same property?