Tonight after my wife got home, I removed the sliding door. Bear in mind, I had already removed the cover for the inside door handle, the upper window trim piece, and the inner door panel.
1. The white plastic panel that rolls up the wire for the power windows had to be removed. There were five 10mm shoulder bolts that held it in place.

You also need to disconnect the connectors for the door lock and window switches. Do this carefully by hand or with a small screwdriver.
2. The clips that attach the wiring for the power windows and locks had to be removed. With a trim removal tool, I carefully pried 4 of them off of the plastic inner door panel. Once they were all out, the white piece with it's wiring can be placed inside the car.
3. I marked and removed the bolts/brackets to pull the door off of the car with a silver sharpie. (My wife's Mazda5 is black.) Now for this step, I removed the two 12mm bolts from the front lower slide assembly, leaving the arm attached to the door. I moved the front lower slider forward out of the way and put a rag around the arm so it could still rest and slide a little bit on the door sill if it needed to.

There were also two 12mm bolts on the rear of the door, along with a 12mm nut. I removed the two bolts from the rear of the door completely, but I saved the nut on the back slider for second last as it has a stud that is useful for holding the door up. It can be loosened though, so you can easily twist it off by hand when you are ready to remove the door.

I also left one of the two 10mm top bolts in the arm that holds the upper support for the top slider in. I had my wife remove the top bolt while I held the door (you definitely need two people to do this).

I picked up the door and set it down on a tarp that I had folded up on the floor beneath the door. The door isn't too heavy, but it is awkward to carry. I would guess it weighs about 40 lbs.
4. We put a couple of old blankets on the floor of the kitchen and I brought the door inside and set it down with the inside facing up.

Then I carefully removed the inside handle. I think there were 2 10mm bolts that hold it and one or two #2 phillips screws.

There is a rod that goes to it that you can rotate the handle off of to remove it from the door.
5. Next was the little cable that attaches to the green connector for the handle mechanism.

I carefully pried it and rotated it free, and afterward I pried the cable out of it's plastic hold down notch.

Then I was able to remove the handle mechanism and rotate it up out of the way still attached to the cables. I was careful not to stress the cables, and I put a few paper towels down on top of the window glass before I set it down to avoid scratching the window or it's tint.
6. The window motor was next. There were three phillips/10mm screws that held it in place. Once they were backed out, I was able to pull the motor assembly out of the regulator cam.
7. I had mistakenly thought that the large black plastic panel was sealed with that strip of tar-like rubber that you usually have to heat up or remove on a hot day to get it to come off. It is actually a regular type of rubber gasket, which is nice. On this panel, there are numerous 10mm bolts that hold it down.

I removed all of them, including the two 10mm nuts that hold the window track in place. There was also one odd plastic rivet that I used the trim tool to remove.

Once I did so, I was able to push the cable through the plastic panel, and carefully pry it up to remove it from the metal of the door. That got me to the point where I could troubleshoot what was going on with the door handle.
First thing I noticed was that on the front part of the part where the handle goes through the door, there was a broken ear on the plastic that holds the die cast pivot for the latch rod.

That rod goes down to a pivot that attaches to the cable with the green connector that I had had to work through the large black plastic panel I had just removed. One side of the pivot has a roller/weight attached to it. At first, I also did not see the latch rod or its pivot at all because it had swung down inside the door area below where I could see it. I found it and rotated it up to see how everything actually works. I made a video and I will post it on youtube kind of showing how the thing operates in case anyone else has this issue.
https://www.youtube.com/watch?v=Y3Bj6sibxoM This raised a big question for me though, as I noticed my latch rod has a pretty significant bend to it. I am curious if it is supposed to be totally straight or not. My wife mentioned having some issues with it releasing even when it was nice outside after the door had been locked. I see also that many of the pivots on the main handle assembly are really sloppy and worn out. I think I will be trying to replace that whole mess with a newer, tighter assembly. I am also going to replace the belt molding for the door's window because I noticed it has many cracks that water could be getting down inside the door through. So there you have it, with pics... The install is the reverse of removal. You can actually do it by yourself if you take your time and are careful.