Yes, you can. Categories can be created just like any other article, no matter that the red lettering says there is an error. That just means the category has not yet been created. Creating a new category is an important decision, however, and probably one you would want to discuss with the community before undertaking.
Only articles, images, lists (etc.) from and about canon sources may be added to BattleTechWiki, right?
Kinda sorta. BattleTechWiki has a policy regarding fan fiction (or 'fanon') that separates alternate subjects written by fans from the official and canon subjects on which BTW is focused. However, if an item of fanon has influenced the canon material (in some form or function and outside the scope of BTW), articles regarding that fanon product may be included here. Likewise, sometimes graphics that do not exist (in any form) in the canonicity but are described in it may be generated and included here, specifically as a means to assist in the understanding of the subject material. Modifications of official products, such as images cropped to focus on the subject matter, are also allowed, when properly sourced. Art for art's sake, however, is not allowed.
Do I need to register in order to use or edit the wiki?
Nope, not at all. You're free to make changes as you see fit, just as any other User may make changes to articles on which you have participated. However, there are some minor benefits to registering: the ads at the top disappear, you get recognition for the work you've done, and the Word of Blake will be able to add you to their database (only for spam purposes, we assure you).*
*Seriously, your address will not leave the site. Any e-mails sent to you via the site (if you enable the option) will be blind (i.e., the sender will never see your address).
I've found a page that has been obviously vandalized. What should I do?
You can always call the article to an Admin's attention, by leaving a note on this page, especially if you have some doubts. But, if its obviously vandalism, Be Bold and revert it yourself. Click on the history tab and locate the last edit made before the vandal struck. Click on the time/date link before the good editor's name, which will open up the last clean version of the article. Click on the edit tab, write "revert:" (or just rv:) and a reason why into the summary field. Save the page and you've now corrected the vandalism. Thanks!