1) The other day I was out driving, and out of curiosity, I pushed in the "RR Diff Lock" button to see what it would do. (I know, I probably should have read the owners manual first, but I didn't). Anyway, a blinking red light came on in the dash. I don't know if it was my imagination, but it actually felt like the truck's front wheels were no longer engaged, almost like it was running in rear wheel drive only. Is that possible? When I read the owners manual it says the RR Diff Lock light, when blinking, means it is not working.
As others have explained, you have to be in 4L for the diff lock to engage. And be happy that it didn't, because having the differential locker engaged on pavement for more than a trivial amount of time would have destroyed your rear differential.
Based on this I suspect that this is your first 4x4, or at least the first one with a locker. You might check the regional sub-forum for Atlanta or see if there's a Facebook page for local FJ owners, and go on a few offroading runs with them to learn what your truck can do, and when it's advisable to try to use some of the features of your truck.
2) This is probably real expensive, but if I wanted to take my FJ to a mechanic and have them modify it to give me the option of being in 2WD or 4WD, would that even be possible? If so, does anyone have an idea how much it would cost? I live in the Atlanta, GA area.
It should be possible to swap in the differential from an automatic transmission FJ, but it's doubtful that anyone has ever done it. The people who buy the manual transmission FJ's typically prefer driving them to automatics, and/or buy the full-time AWD intentionally because they live in a snow state and don't want to have to engage the 2H -> 4H in bad weather.
It would probably be cheaper to sell the manual FJ and buying an automatic. Do you ever plan on offroading it? The automatic is easier to drive offroad. My FJ is an auto because I offroad it. When I want to enjoy driving a manual, I have a convertible in the garage for that. The FJ isn't a fast, fun car that really justifies having a stick.
Good luck! And welcome to the FJ world.