I have several flash units. Both the shoe-mount type and the "studio" type. The portrait has to be taken in a room that happens to not have windows. It is always dark in there. I can bring a light just bright enough to help me see the subject, but the camera needs some light to focus. I don't want the warm cast of any incandescent lights. Do I let the TTL system in the camera take care of the exposure? Do I have to focus manually, without any other option?
TTL flash exposure typically requires that all flashes are compatible with a single TTL system, and it's not clear from your description which if any of your flashes would work with TTL. Manual flash exposure is a bit more work, but is will work with virtually any flashes. If you don't know how to perform manual flash exposure, Strobing 101 is a great place to start.
Studio strobes often include a "modeling light", which is a small, always on, incandescent light whose brightness is scaled to the strobe brightness. This allows you to easily preview the effect of the strobe without taking a picture, and also provides enough light for autofocus or manual focus.
Many cameras, include the Canon T2i have an AF assist lighting system, that when enabled allow the camera to achieve autofocus, even under ambient lighting conditions that are too dim for effective autofocus.
Finally, if you use a bright flash combined with low ISO and your maximum flash sync shutter speed, you should be able to have ambient light that is bright enough to focus with, but that will not contribute significantly to the final exposure. You can confirm this by taking a picture with these ISO, shutter, and aperture settings and with the flash temporarily disabled. If ambient lighting still shows up in this exposure, reduce the amount of ambient light, reduce the aperture, or lower the ISO until you achieve low enough ambient lighting that you can still focus by, but does not contribute significantly to your exposure with the flash disabled.
Use only a modest amount of constant light, just enough for the focus system to work, and use the fastest X-sync speed your shutter has. That will make the ambient light during the short exposure interval so dim relative to the flash as to be irrelevant in most cases.
For example, to properly expose a picture in a room lit just enough for the auto focus to work might take 1/30 second at f/2.8 (just picking something vaguely in the plausible range, you'll have to measure your setup yourself) and some convenient ISO. With all the stobes firing, you might stop down the lens to f/11 and whatever your fastest X-sync speed is, let's say 1/250 second. Compared to the ambient light exposure, you're 4 f-stops down due to aperture, and another 3 due to shutter, for a total of over 7 f-stops the ambient light will be dimmer than the strobes. At that point, the ambient light is basically irrelevant.
I can bring a light just bright enough to help me see the subject, but the camera needs some light to focus. I don't want the warm cast of any incandescent lights.
You're worrying about a problem that doesn't exist. The flashes of light produced by even a small hotshoe flash gun are so much brighter than standard interior lighting that provided you shoot at or close to your sync speed (e.g. 1/250s) the room lights simply wont show up - you can safely leave them on for the whole shoot so you should have no problems focusing.
You can test this by setting your camera up for flash, say 1/250s f/8 ISO100 and taking a shot with just the room lights on. It should come out black. Turn the flashes on and if the result is well exposed then you can be sure the room lights are not affecting your shot.
These were shot with all the room lights on. Notice that when the background light is off [left] the background becomes black, the ambient light doesn't touch it, likewise when the background light is on and the foreground light is off [right] the subject is totally black:
TTL metering is fine too - as long as the camera is forced to use a low ISO number (e.g. 100) and fast shutter speed, it will calculate the amount of flash needed effectively ignoring the room light (which wont be contributing enough light for the camera to care about).
I actually prefer to have the room rather dark when doing flash photography. Flashes are much brighter than pretty much any room lights, but the room lights will still give some contribution and take away from the control of the flashes.
If all your flashes are compatible with your camera's TTL system, then you can use that, but unless they are all the same make, there is a reasonable chance they may not be compatible with each other. If they are not, you would have to manually configure the power (and you will still want to manually configure the power ratios between your groups).
As for focusing, most flashes will either offer a modeling mode (where the flash fires repeatedly for a few seconds) to allow for seeing how the lighting looks and for focusing to occur) or more advanced models may have an assist beam that can be projected on to the scene in order to further aid focusing. You can also use the modeling mode for manual focusing if you prefer, though generally you want to avoid using the modeling flash too much as rapid repeated use will cause the flash bulbs to accumulate heat which is bad for their longevity.
Use modeling light. Or just turn on the lighting in the room :-) If that is inconvenient, there are cheap Zigbee wireless remotes for switching on and off a 220 V device plugged in. Best if you have a fast-turn-on LED lamp.
Then you focus with available, constant light, and then turn off main light, make the shot with flashes, and turn on light.
If your camera emits pre-flash, it will calibrate the flash power and adjust exposure. Typical for Canon-s. However, with flash-es, you usually go all-manual in a studio, so you will do a few shots, adjust camera settings, and that's it.
You don't have to focus manually (though you may choose to if you wish); if your flashes don't have modelling lights, you can easily use a small torch (flashlight) to put enough light onto the subject for the camera to focus and turn it off (or point it away) before tripping the shutter.