Should we expect Leipzig Glossing Rules to work for all invented languages?
No, not all of them, but the set of languages for which they will not work is exceedingly restricted. Basically, just ask yourself "can my language be written in a primarily linear format?" If so, the Leipzig Glossing Rules are for you. If it is carried by sound, you can gloss it. That right there covers very nearly every language that has been asked about on this site so far. If it's a manual sign language, there's still an exceedingly good chance that you can gloss it. If it's a fully-2D non-linear writing system, the alterations you will have to make to the conventions to make it work then become sufficiently large that it would no longer make sense to call the result "Leipzig glossing". But those aren't exactly common.
How does one handle grammar that may not match up with the "standard abbreviations"? Lojban comes to mind, since they as a community reject that grammatical classes in Lojban have any counterpart in natural languages.
Exactly the same way the professionals do: by providing a glossary of your non-standard abbreviations.
Conlang audiences are just as likely to be an amateur audience, unlike the readers of linguistics academic journal. Is there a better way to gloss for a popular audience?
That depends on what you are trying to communicate to that popular audience. In some cases, a full gloss may not be needed, even for a professional audience. A simple set of parallel translations demonstrating the effect of a straightforward morphological or syntactic alternation, for example, may be sufficient for its purpose. But in the general case... I don't think so. Leipzig glosses are intended to be easy to understand, and they succeed at that about as well as it is possible to do in most cases.