A dvd player is normally restricted to dvd resolutions. This means that the resolution of your video file can be no bigger than 720 x 480 (NTSC) or 720 x 576 (PAL). We use NTSC in the US and Canada, but PAL standards are more common around the world.
It doesn't matter that your dvd player can upscale to 1080p resolutions. The input file has to be within normal dvd boundaries. That is, unless you have a truly exceptional dvd player. I don't know of any off the top of my head that can use higher resolution input files.
Blu ray players are a completely different kettle of fish of course. They are designed for larger resolutions.
Now, on to your other question. Blu ray movies of 1080p resolution are normally anywhere from 18 to 40gb in size (movie only). The H264 codec can do some amazing things in compression, and it is possible to fit a 2 hr movie at 1080p onto a single layer dvd, but you cannot do this without some loss of quality. Converting to 720p to fit onto such a small size disk will help, but even it will not be comparable to the original blu ray movie.
If you want to try to compress a regular blu ray movie to such a small size, I recommend a program called BD Rebuilder. There is a guide to using it in my signature. If you are starting with mkv files or mp4 files, you'll need a different program....something like Ripbot264 or Handbrake.
And you must realize, the high def files will not play on the vast majority of dvd players.
You can of course convert to regular dvd video using AVStoDVD, but that will not be high def.