To clarify most things in one post, hopefully:
Yes, Opera Mini is free, as is the other mobile browser from Opera, Opera Mobile.
Neither support Flash, though typically that is not too much of a problem, considering there are specific Youtube applications for devices anyway and other Flash content on websites is advertising. I will admit there are some sites where Flash is used for useful purposes and the lack of support is annoying, but don't forget mobile browsers are used in addition to a desktop browser, not instead of.
Finally, the differences between Opera Mini and Mobile. The most obvious is that Opera Mini runs on just about any phone, whereas Mobile does not and has a limited phone support. This is because of the major difference between them; Opera Mini sends everything to and from the Opera servers (and having used Mini for many months, it does speed up a slow internet connection and is not at all problematic) and so requires less resources on the host mobile. Opera Mobile instead does all the processing on the mobile, but like the desktop browser, Opera Turbo can be enabled.
Hope that clears a few things up.
/sent using Opera Mobile from a Nokia E63