You say you want math-specific answers, but there is nothing math-specific in your question. At any rate, your question got many upvotes, so I am throwing in a quick answer even if it is not appropriate.

1. You can certainly send more than what employer institutions asks for, but I would advise you against it. One or two extra letters of recommendation might help, but sending other types of documents than asked for will probably not look good on you. In all your application, try to be concise and to the point, and bear in mind that it is not an essay contest. If there is no size limit on the research statement, then there is no point in preparing a short version and a long version.

2. A publication list section in CV simply means what it says: a list of all your relevant publications. Abstracts should not be included.

3. Probably the letters are the most important factor in the first filtering. So choose your recommenders carefully: the more the person knows you and is known by others, the better.

I was writing this very quickly, but hopefully it has some value and use.