You may wish to consider an S-corp for a business that generates short term gains/ordinary income.  

It has two unpermitted Enclosed patios, a unpermitted aluminum patio and a unpermitted fiberglass pool.

If you did it right, and refinanced out but still own the asset, that's the ideal situation, but finding opportunity to do that today is something "ordinary" investors are really struggling to do, if they're able to do it at all.

The article mentions taxes gained and lost - is this the place to factor in taxes paid on the income from the property (is that taxed at a rate of ordinary income?)

I was originally planning on jackhammering it out myself, hiring a dumpster, and replacing the structure with a basic 10' x 10' paver patio and seeding the remaining ground for next season.

It's not cap gains, flipping is ordinary income tax.

The tenant talked to the neighbors, and the neighbors said that the HOA does not allow smoking in the common areas or on the patio, so they smoke inside their unit.

As a result our flip projects will be recognized as ordinary income. 

I'm assuming you have some kind of concrete patio under the tile.

We have beautiful weather here in So Cal, so maybe we can make this patio-like.Your ideas and advice would be appreciated!