Do most Search engine robots stop reading content on a page when it
hits a </body> or </html> tag? If no, does it read content until there
are no charters left to read before moving on? Suppose I have several
beginning and ending tags on one page would this confuse a robot
reading the page or would it still be able to index the page properly?

Interesting question. I'd be curious to learn why you want to know
this. But regardless... The best way to answer SEO questions is to try
it yourself. If you already have a page that is indexed in Google,
just create a page that meets the conditions you described, upload it,
and link to it from a page that is already indexed by the search
engines. Wait a few days, and then look it up in the search engines to
see what got indexed.

The bots will read text beyond the tags, but not WEIGHT that content
at all (in other words, they see it, but it doesn't count for
anything).  Google bots even ignore meta tags at this point, and
really only index your site by content found in the BODY of the HTML
document.