TextMate is only showing some bundles, and not highlighting the code anymore

That's probably because the PDF uses actual curly quotation marks (which are indeed syntactically incorrect). The most straightforward thing to do would be performing a search and replace on the copied text in your text-editor.

According to the 12.6 user manual (p104), BBEdit offers code completion based on:

• clippings (both language-specific and universal);
• ctags symbols computed by running the current document through 'ctags';
• ctags symbols found in 'tags' files in the current document's hierarchy;
• predefined names for the source code language at the point of completion;
• language-specific completions (both predefined and derived from the current document’s content);
• dictionary words provided by the system spelling service

By default this completion is triggered by F5 but "You can change this equivalent via the Menus & Shortcuts preference panel."

(And, yes, Kite support was deactivated in version 12.1.5.)