Does Ubuntu now release 64-bit and 32-bit under the same ISO?
No, they are separate images. You should select the version that you want from the download page. On the page you linked to, amd64 denotes the iso for 64bit, and those with i386 are 32 bit iso files.
You should also get the md5sum for the iso that you download and check that is correct before burning the image to a CD/DVD or USB stick.
If you have an unreliable internet connection then you might want to get the files from one of the official BitTorrent downloads because it will have extra error checking and the ability to recover and restart.