Congrats! The T3100 was my first ever vintage Toshiba portable, it's an awesome little machine.
1. As far as I know, there are no 720K floppy images of DOS 6.22, but I could be wrong. It can be manually installed, but I wouldn't be able to provide the exact steps off the top of my head as my T3100 came with DOS 5.0 on it and I'm content leaving that as-is because of its smaller drive space footprint. I know at the very least it would involve reformatting some of your blank floppy disks as 720K (You'd have to cover the HD hole with masking or electrical tape to do that) and manually copying files off the DOS install disks onto them, as well as creating a 720K 6.22 boot disk with FDISK and FORMAT on it to partition the hard drive and make it bootable.
2. The T3100 doesn't need a park command. The built-in hard drive automatically parks after five seconds of inactivity. Just make sure the "Left Disk" light isn't lit, and you should be good to turn it off. Beyond that, it can be treated just like any other drive. Be aware though, depending on how long it's been siting unused, one thing you MIGHT need to do before partitioning the drive is perform a low-level format. This was a common routine maintenance task that sometimes needed to be done to early hard drives from before the introduction of IDE, as gradually over time the platters on pre-IDE drives can literally begin to lose their ability to hold a magnetic charge. Low-level formatting a drive will write over every single sector on the entire drive, and basically "refresh" them so they can be used again. This is also a good way to permanently erase any personal data that may still be on the drive, depending on where the computer originally came from. This is not something the regular DOS FORMAT command does, you need a special program compatible with the installed model of hard drive to do the low-level format. After that, you can then run FDISK and then a do regular FORMAT to prepare the drive for use. My answer to question #4 below has a link to a diagnostic disk that will allow you to do a low-level format.
3. I would not know the answer to this question off-hand since the oldest DOS game I've ever played is DOOM. One thing to keep in mind though, the T3100 has CGA graphics, but the internal screen is monochrome only, not even different shades of orange. It's either orange or black, so even if you can find CGA games that will run fine on that system, they likely may not look good on the internal screen. Luckily you can connect an external CGA monitor to it just fine. There's a keyboard command to switch between internal and external screens. I don't recall it off the top of my head, but the technical manual should have that info.
4. Cannot think of any demos, but for software I'd recommend you grab the Toshiba diagnostics disk image and write it to a 720K-formatted floppy using Rawrite or WinImage. It can be downloaded at FOUND! - Toshiba CE Diagnostics Disks. This'll let you run various diagnostics on the system to make sure everything is working properly. It'll also let you low-level format the hard drive, which I'd definitely recommend you do before trying to install DOS. If after the POST "Error in CMOS" screen pressing F1 doesn't take you into the CMOS setup, then you can also use this disk to get into the setup and change the settings if needed.
5. BIOS battery can be replaced. Although it is possible to purchase a compatible replacement battery, the connector on the end of the new battery wiring will be different, so you'll have to cut the connector off both batteries then wire the old connector to the new battery. Be aware, the T3100 is A PAIN to open and work inside. The wiring connecting the screen to the motherboard is very short and permanently attached, there are no connectors that can be unhooked. The best you can do after unscrewing the case is lift the top up like the hood of a car at most 45 degrees, then you'll have to use something to prop it up so you can have both hands free. Then, when you go to put it back together, you need to make absolutely sure the little spring-loaded plunger sticking up from the motherboard hooks correctly back into the screen hinge (Open the screen HALFWAY before you fold the top of the case back down, then close the screen once the top is all the way down). That plunger is the switch that turns the screen on and off when you fold it open or closed, and is easy to break if you don't get it hooked back into the hinge correctly. If it breaks then the screen will stay off when you open it, essentially rendering the system useless.
As for the screen, columns of pixels that are either dead or stay on are unfortunately common on the early Toshiba plasma-screen portables, my own T3100 has a line running down the center that is always on. I do not know of any easy way to fix this, as doing such is getting outside my area of expertise.
Some other tips:
-Make sure the "A/B/PRT" switch on the left side is set to "PRT", this way the internal floppy drive will be drive A: and the printer port will work normally. The other two settings are for use only when you have an external 5.25in floppy drive connected to the parallel port, and are used to tell the system which drive letter the external floppy drive should use. If you don't have the external floppy drive for the T3100, then there's no need to change it from the PRT setting.
-By default, all six of the DIP switches on the back below the power plug should be off (Down). The only one you might potentially need to change is switch #4, which changes the printer port to bi-directional...this is good if you want to connect a parallel-port Zip drive or use a parallel LapLink/Interlnk cable. Switch #5 can also be changed if you absolutely want to, all it'll do is change the on-screen text between standard and bold print...really down to personal preference (I personally find it easier to read in standard mode).
-Don't leave the screen on a static image for too long. Plasma screens burn in VERY easily. After installing DOS, it might be a good idea to look for a TSR screen saver or screen blanker that automatically kicks in after a few minutes of inactivity...or just close the screen when you step away from it to turn it off.
-If you are absolutely curious...yes, it will run Windows. I'd recommend no higher than Windows 2.03, but if you're feeling adventurous it WILL run Windows 3.0 (But not 3.1). It's a very tight squeeze on a 10MB hard drive though, and you have to use a hex editor on the "Olivetti" video driver files to get 640x400 resolution working correctly, otherwise you're limited to standard CGA resolution.