MediaCoder

Could anyone please give me the best settings for converting a 1920 x 1080p, 2h:10M:05s full length DVD movie to 1GB MP4 format without changing the dimension (pixels).

I have just separated your post from where you placed it (not relevant to that Topic and unlikely to get a response).

It is very easy to maintain the dimensions, called Resize on the Picture tab.
Just, uncheck (click to remove the tick) in the boxes beside "Resize" and "Framerate".
I suggest you begin with the settings suggested in the Encoding Guide, listed under Tips & Guides (link below) and maybe increase the Video bitrate it any loss in quality at 1000 kbps is noticeable on the viewing device.

You could achieve the requested 1 GB output file if you chose video at 1000 kbps, assuming the audio is at around 80 to 160 kbps, say.

I have just prepared a Guide relating bitrates with file size.
So, we may as well use it!

To achieve the requested 1 GB file size for the 2h 10 min movie, use equation 2 in the Guide (viewtopic.php?f=17&t=12438). For example, with S = 1024 MB (1 GB) and T (duration in Min) = 2.167 x 60, we have that V+A = 1101 kbps.
Hence, with the video bitrate V at 1000 kbps, the audio bitrate would need to be at A = 100 kbps. Of course, the latter needs to be a multiple of 8, such 96 kbps.

At 1000 Kbps , 1080p won't look good, even if you you encode with x264 using preset placebo.
Using preset veryslow instead will be much faster and only decrease quality by 1%-2%
Exception being that if the video is animated, then even 500Kbps might not be too bad.

If it's a 'normal' movie, you're probably going to want to degrain/denoise it first.