Modchips
Retail:
All are at about the same level (now that Wiikey has had it's update), they can play all Wii backups and all (in general) Gamecube backups of the same region as the console. Multi region support/on the fly patching is enabled on most chips (except very early Wiinja models) so the vast majority (if not 100%) of other region Gamecube games work and a percentage of Wii games work. There is a wiki available on this site to give you an idea of cross region compatibility.
Do-it-yourself/Open Source:
There are a number of open source mods available at the moment whereby you flash the code to a blank chip. Advantage of these is that they're always being worked on and are free, disadvantage is they currently (I may be wrong/out of date here) lack some of the features of retail chips. You also largely need some knowledge/equipment for the flashing of the chips.
Firmware Updates:
Firmware updates are currently safe if from the same region. Updating your console via the internet is safe, as is updating your console via a backup from the same region. If you accept and update from a title from a different region it will brick or semi brick your wii (Semi-brick = will boot games but not allow you access to settings etc). There is a tool doing the rounds called "Brickblocker" which, for every title except Metroid Prime 3, currently works. It strips the update from the disc thus eliminating the problem. If run on Metroid Prime 3 (and presumably newer games containing an update) the game will just refuse to boot.
Some updates (Not full firmware) such as the update on Metroid Prime 3 won't brick your console. They will however add duplicate channels which are unusable (Weather and News I believe). The current "trick" for getting around this is to unplug the console when the update is at 12% or so, no duplicate channel and then a brickblock'd version of the game will run.
In general, to be on the safe side, don't run other region games unless you have an idea what you're letting yourself in for.
Online Games:
No-One thus far has had a problem, but I imagine it wouldn't be greatly difficult if piracy increases for Nintendo to put in a check similar to that used on Xbox Live to see if the user is running a DVDr rather than original. Booting an original game and playing online with a modded console should, I imagine, be fine.
Creating Your Own Backups:
There are several guides to this, but it's not really the easiest of processes. As far as I'm aware (It's not something I've looked in to much) you need a specific DVD-ROM drive on your PC, flashed with a specific firmware. Dumps currently take in excess of an hour to make. There is a tool to dump a disc image to a memory card FROM the Wii but I think there's either a file size limit, or the Wii is limited to 2gb Flash cards.
Types of Wii:
There are currently a number of different models doing the rounds. Some of the newer ones have had the legs you normally solder the chip to cut-off. To mod these you need to grind the chip down till you can see the points and solder there. I also believe there's a new drive-chip which is currently un-moddable.
Hope all that helps.