I have been requesting information about Check Point's regular expression engine for a few years.
The match is definitely case-sensitive. There was an option to make it case-insensitive back in the R77 family, but it seems to have been removed in R80 and up. Plan for lots of [cC][hH][aA][rR][aA][cC][tT][eE][rR] [aA][lL][tT][eE][rR][nN][aA][tT][iI][oO][nN].
While I know it seems to construct an entire URL including scheme and path, I don't yet know if there is any normalization performed when constructing the URL. For example, is the name always lowercase? No idea. Domain names are case-insensitive, but paths are case-sensitive, so this kind of implementation detail determines how we can write expressions.
The match is definitely run against the whole URL including path, as when a coworker tried to use "\.[aA][rR]" to block access to Argentinian domains, it caught files such as "right.arrow.png" on otherwise allowed sites.