You actually can copy/paste from the web if you set up your multibyte settings correctly and either use font-linking or a font that supports the characters. Also, the next mIRC is supposed to include full unicode support, so you'll be able to copy/paste even more easily at that point.
For multibyte settings, these work fine for me:
Alt-O > IRC > Messages
Check multibyte editbox, UTF-8 display, Font linking, and ANSI codes.
Note that there may be other combinations that also work, but that works fine for copy/paste from a browser.