I'm intrigued, but ...
* I don't want to find myself out in the middle of nowhere with bike mechanical issues (I am inept)
Not often a problem. One solution is a cell phone and the phone number of anyone back home. I haven't had mechanical issues, but have had my wife come fetch me on two or three occasions when things didn't go according to plan.
* I am a huge fan of clean indoor plumbing
On the Lone Star Randonneur routes, the controls are generally convenience stores around 30 miles apart, or if the controls are farther apart, there's usually stores you can stop at in between. The rest room facilities are usually on par with what you'll find at most other businesses. Meaning, unless you just refuse to use anything but your own bathroom, you're probably going to be okay.
* I am afraid of being out in the middle of nowhere running out of drinking water
Solution: Plan ahead. Rest assured, none of the rest of us want to be in that situation either. You will generally always know where the next water is. What you won't always know is how long it will take you to get there and how much you'll be drinking as you go.
* Sleep deprivation makes me more prone to seizures and other neuro issues.
Generally, a 200k ride will take 8-12 hours depending on your speed and fitness, so sleep deprivation shouldn't be an issue there. A 300k might take up to 18 hours or so, but that's still more or less normal waking hours for most people. Rides over that length may get into sleep deprivation, but if you're just starting, that's not a concern.
* Most riders having my slow speed level aren't interested in riding more than 20 miles
What exactly is your slow speed? LSR does have some slower riders, some in the middle, some that are fast. One challenge they've been working on for the slower riders is trying to get them all at the same ride. Anyway, it's hard to say how you fit in until you've been out there to see the pace.
* Still working on improving my physical conditioning
Ditto. And let me tell you, riding 130 miles is a heck of a way to work on it.
Speeds are pretty relative. There are people who will do a 100 miles, average 18 mph, and feel like they were slow, whereas I've never managed to be that fast.
You may want to check into the Fort Worth Bicycle Association, I think it is. Some of the people there are also in LSR. They have a bunch of organized rides of different lengths. Charity rides are another good way to see how you're comparing to some of the other riders around.
When I started out in randonneuring, I wasn't sure if I could make the cut-off times or not (I was riding my Worksman single-speed cruiser). So before I did a 200k route, I did 100k and 160k permanents by myself to see how it worked, and then went on to the 200k. I had previously done the HnH 100 and a number of 100k charity rides as well.