The basic problem starting the trouble is a fairly usual one with Debian package management and incompatible Debian variant releases. The .deb package install scripts may suggest or even perform very harmful things.
I doubt it will be worthwile to try to patch the USB install - the moral is, always make a backup copy of such installs, and update the backups as appropriate.
Try using Synaptic package manager - that one has worked well for me, but always update package lists.
And if a package you are looking for is not in any of the ordinary repos, it's good practice to check in advance that it will actually run on the actual combination of hardware and Debian variant.
I would not say installing new packages will always cause difficulties, but I don't consider it particularly smart to start with the CD release of Knoppix if you are going to install substantial amounts of extra packages. Start with the DVD instead, that's much more likely to have necessary infrastructure properly installed.

And forget about remastering until you don't consider yourself a Knoppix newbie any longer.

As for Debian and Opera, Google tells me:
https://wiki.debian.org/Opera‎
Oct 30, 2012 - The recommended way to install opera, is to add Opera's package repository in your /etc/apt/sources.list.