My site is http://thaihealingmassage.com. I am running wp 3.5 and ATA 3.7.3
I have NextGEN Gallery installed and it only shows images on a page called "photos". However there is a huge amount of NextGEN Gallery code in the source code of my home page and every other page.
I cannot imagine that this is very good for my page load speed and also not for the Google crawl spiders.

Is there some way to fix this with NextGEN Gallery, or is there another, less code intensive way to show a bunch of pictures on one page which get bigger when you click on them?

Try swapping to twenty twelve and see if the code shows up then. If it does, you should contact the plugin's developer for help