1) There is no real background check when it comes to this... ie. they aren't going to require a swab of your mouth or urine sample. Haha - that would be an interesting law school application... gross!
This is really just based on the honor system. Now, I have no idea what the bar does - but I'd imagine they will do some sort of basic background check. They will cross reference information from this with your law school app. Again, I'm not sure if they go into detail of your heritage though.
2) That said, the general consensus is if you typically mark "hispanic" or "mexican" or something similar on random forms, then go ahead and do it here. If not, tread lightly. You could be viewed as a stinky person who plays the system (again - I don't think anyone would ever even be able to find this kind of information - but I'm sure there's some ethical dilemma here).
3) My opinion... your mother is almost entirely from a mexican lineage - period. This is close enough for me to consider your heritage significantly Mexican (you are basically 50% Mexican!). And whether you realize it or not, the journey that your immediate ancestors took to get to you is a unique and, more than likely, difficult one. Moreover, your specific ancestry (Mexican) is one that our social structure, for whatever reason, has significantly alienated from the legal profession. This is precisely what law school are looking to correct for (and also be able to boast about their high diversity population - but thats besides the point). So... yes, mark URM-Hispanic/Mexican status!