Date substitutions return the current day. Date fields return whatever the user specified. If you want it to trigger on the date of then just to a straight Equal To. You can't do a date for tomorrow because the date substitution will always return the current date.

Example:
[cb:date format="Y-m-d" /] Equal To [cb_date]
[cb:date format="Y-m-d" /] = 02-02-2012
[cb_date] = 02-02-2012
Condition: Passed

[cb:date format="Y-m-d" /] Equal To [cb_date]
[cb:date format="Y-m-d" /] = 02-02-2012
[cb_date] = 02/02/2012
Condition: Failed

They have to match exactly, same format and everything. You can try "Greater Than" or "Less Than", but again no idea if that'll work.

I don't know what else to tell you. I've DIFFed 2.2.3 to 2.3.0 and there is absolutely no changes in how conditions are handled. The only change was I added parsing of additional trigger variables for substitution usage, which doesn't even affect conditionals (being fixed for 2.3.1).

My guess is less than and greater than aren't working at all. Computing the less than and greater than of a string is a slow process and unreliable for PHP. when it's not a clear cut typecast. For example if it was just numbers then less than or greater than would be fine but as there is formatting (-) it doesn't really know how to handle it (PHP can't just assume it's a date).

Really strtotime needs to be applied to both before doing less than or greater than. There's no real date condition handling in place, you need basically custom conditions for dates.

I've plans to add some sort of formatting rules to the conditions, but no idea when. Which would allow you to do strtotime or trim or other PHP functions on the values.

No idea, it will require several additions to the conditional usage as well as figuring out how to properly display it for configuration. I've plans to add formatting feature, but again no idea when.