For the TI-92, there is only major kernel: Fargo II. This was developed by David Ellsworth, and the current version (which seems to have no problems with it) was released nearly three years ago.
Installing Fargo II requires you to send a backup file to your calculator. This is necessary because there is no built-in assembly support, and thus a backup must be sent to overwrite system memory areas to trick the calculator into executing the custom machine code.
Unfortunately, Fargo II must be sent to the calculator from a computer, not between calculators. Once Fargo II is installed, programs for it can be sent between calculators. Additionally, the backup must be made specifically for your calculator's ROM version. The Fargo II package includes a program which will automatically get a backup from your calculator, install Fargo II in it, and then re-send it to the calculator. You can also get pre-built backups from ticalc.org. Note that installing Fargo II will erase all programs currently on the calculator, so you should make copies of anything you need to keep first.
It is not necessary to have a "shell" to run Fargo II programs; you can just run them from the home screen with commands like "thisgame()" for a program called "thisgame". You can use shells, however. To do this, press ON and then + to start a program named "shell". Fargo II is distributed with a shell called FBrowser which will display a list of programs on the screen and let you choose from them. It also allows you to go through the various folders on the calculator.
It is, however, probably a better idea to use the "FPL" (Fargo Program Launcher) shell which is available on ticalc.org, as this shell also provides limited "crash protection"; that is, you may be able to escape if a program has crashed. Note that this crash protection is far from completely reliable, and if you do recover from a crash some data may be corrupted. For this reason, if you do have a crash, you should reset your calculator and replace all programs with original, non-corruped versions as soon as possible (of course, be sure to make copies of any new data you have on the calculator that you need to keep).
Many Fargo II programs require additional files called "libraries" to run. To get them to work you must also transfer the library files to the calculator. The "standard" libraries included with Fargo II are required by a large nubmer of programs, and some programs also require other libraries (which would usually be included with the program).
There is one other kernel, [tie], which you can install on the TI-92 to run programs made for the TI-92 Plus. This currently requires a converter to be used on the TI-92 Plus programs, and also it does not allow all programs to run.
Yes, it can. You do have to build this support into the Fargo II kernel, however, as it is not there by default. The process of doing this is simple, and explained in the Fargo II manual. You can also get backups with this support pre-installed from ticalc.org, but you still should read the Fargo II manual to know how to deal with libraries for the older version of Fargo.
Source Guru is a program that I wrote, intended to demonstrate the possibility of automatically converting programs for the TI-85 to run on the 68000-based calclators. It did succeed in that purpose, as it made working conversion of several programs.
However, general users should avoid these programs, as they won't usually run as well as programs written directly for the 68000-based calculators, and also currently have some compatibility problems.
This means that a program has crashed. It will be necessary to reset the calculator to use it further. For more information, see the FAQ section on crashes which tells you how to recover from them.