I can answer some of your questions.
Quote Originally Posted by SnowGrow
I'm planning to grow one plant about 20" tall after the flowering is done. I'm hoping to get about 2 more grow boxes like this. I ordered 5 feminized white widow seeds.
Why two more boxes? Isn't it better to use all together as one unit. Good for lighting, air, fitting.

-Seedling and through vegging phases are about 20 hours of light and 4 hours dark right?
You can do with 24/0, 20/4 (mostly for auto flowers), 18/6 (This is most people do)

-Flowering is 12 of light and 12 of dark right? and you're supposed to keep the starting hour of night the same when you switch times to minimize stress right?
Yes, 12/12. Normally, when the last vegging day ended, switch the timer for 12/12. 12 hours later flowering day will start. If you need to change starting hours just make sure you don't give a long day. You can give a long night. Even up to 36 hours or 48h (I think, I don't remember well) to tell the plant "flowering period started".

-How much is it going to grow during the flowering phase? I need to know so that it can get done right at about 20".
Depends on lights. 1 - 1,5 size normally, with your bulb this can be more than 2 size. You can train the plant to keep compact.

Last odd questions, I know plants use CO2 during the calvin cycle IIRC high school bio correctly. Would using bubbly (CO2 filled) water have any advantage over using tap water?
You don't need to worry about CO2 with your setup.

EDIT - BTW. it's a 13"x8"x22" grow box. I'm assuming that's going to be about just right for one plant right?
8" space is narrow for a plant. When i looked at the pictures i thought that was the light fixture only
Like i said in the beginning, you can put together aall those boxes to have one proper box. 13"x24"x22" sounds good for two plants.