"Does Google know that VT = Vermont?"
Yes. Try this search: https://www.google.com/search?q=vt&gl=us&pws=0
Notice on the right side of the screen Google offers information on two topics: Virginia Tech and Vermont. Google knows your query likely pertains to one of these topics but is unsure which, so it offers both.
"Should State names be abbreviated and/or fully spelled out in title tags, meta descriptions and body content?"
In content, you should use whatever format you feel would best connect with your reader. If I was to mention Los Angeles, I would not typically feel the need to mention California since the city is recognized world-wide.
For Miami, there is both Miami, Ohio and Miami, Florida. If I was writing about Miami Ohio, I would definitely mention the state name. If I was writing about Miami Florida, most readers would automatically make the connection to Florida, especially if I mentioned anything to do with the ocean, a major city, etc.
For meta descriptions, most of the above applies but you are once again focused on connecting with your reader. The main goal is to improve click-through-rate for your search listing.
For Title Tags, this should be decided first. The title should be closely aligned, or perfectly matched, to the keyword focus of the article. Otherwise, it again can depend on the situation. Take a look at the following example:
Miami Dolphins start the season 6-0. It is unlikely you would ever wish to add FL or Florida to that tag.
You can perform keyword research to determine which version has more traffic. Many SEOs will make the decision in that manner.
Consider the following exact match local monthly search volume in Google.com:
homes for rent in orlando 2400
homes for rent in orlando fl 1300
homes for rent in orlando florida 260
Clearly, the preference is not to use state information based if your focus is finding the maximum search volume for this query. But what about....
homes for rent in jacksonville 110
homes for rent in jacksonville fl 3600
homes for rent in jacksonville florida 210
In this instance, adding the abbreviated state name to the query is very important.
As with most SEO related questions, the topic can go a bit deep. Keyword research is a critical component to a successful SEO campaign, and takes time, testing and experience to find the optimal variation of the term.
Best of luck.