What is required of a mathematics referee?
My experience as a both reviewer and author in pure math suggests the following:


there is no universal standard for refereeing (some referees check papers thoroughly, some are much more cavalier)
editors may have different expectations/desires of reviewers, depending on the situation and on the editor
the typical expectation is that you assess a paper from several aspects: importance of the paper (this includes factors such as usefulness of the results/methods and novelty), correctness, and presentation/clarity.
refereeing is a process, not a "yes/no" assessment
refereeing well takes a fair amount of time, and results in little personal recognition, but is important for the community


Based on your specific questions, here are my suggestions for how you might go about refereeing a paper:


First assess the results claimed in their general context.  If they do not seem interesting enough for the journal in question, or are clearly wrong, you can simply say that (preferably with more details) and recommend rejection.
Understand the general argument of the paper.  Read the introduction in detail, and then try to understand the main points of the paper.  (You might not read the rest of the paper in order, particularly for long papers.)  Assess the novelty and feasibility of this approach.
If all seems good so far, convince yourself of the correctness of the paper.  This might entail checking all details, or you might be convinced after only checking certain details.  While you should definitely assess the correctness of the result, ultimately the burden of correctness lies with the authors.  Of course it's great if you can check the paper completely, but in my experience most referees don't.  How much effort you put into this may also depend on the situation (e.g., for a completely novel approach with an amazing result, you probably want to put more effort into checking details).
Very often in this process, there will be things you don't understand.  There could be a couple of reasons for this: (i) you aren't an expert in the methods being used, (ii) the authors' explanations are lacking.  In both cases, you can simply ask the authors for more explanations or explicit citations of the results they are using.  It's not necessary to understand everything for a first report, but hopefully you will mostly understand the paper after a successful refereeing process, which may include several revisions.  However, if (i) is a very serious issue, you might consult with the editor.  E.g., if you can read though the details in Parts I and III but Part II is a complete mystery to you, the editor may find someone else to referee Part II.  (Possibly this is a conversation you could have when agreeing to referee the paper.)
In the above process, you'll likely come up with a list of questions/suggestions, both about the mathematics and the writing.  While you should of course include these, these are typically of secondary importance in the referee report (though sometimes if you have some really good suggestions, it can dramatically improve the paper).