Without even knowing your project settings...your workflow involves encoding and re encoding compressed files to compressed files plus generational issues.....so it would be expected that you get a quality hit along the way.
eg mpeg to avi to mpeg
Your source material is delivery format, for watching. You are using it to edit and with each step the quality goes down. If you want to edit, start with editable material from a video camera, not from a DVD.
This is the unfortunate way that it works. The DVD-Video will be compressed and Transcoded to MPEG-2, which is a lossy CODEC. There is no way around this.
Now, you say that you are ripping the files to begin with. Are these already on DVD-Video? If so, they have already undergone one MPEG-2 compression, and then you are adding a second one. It's like making a copy of a document, FAX'ing that to a friend, and then comparing the FAX that he sends back to the original document. Degradation will take place.
Sorry for the bad news.
The thing that was tricky for me to grasp (still is sometimes, as I am new at this editing) is that "what you see " isn't what the program is going to see when you put the file into the program (be it the editor or the DVD burner ) ... The "delivery" format is MEANT to look good....and be small in byte count (compressed) for speed of play and so on....but in fact although it looks good it may not look good at all to the programs...and then to what is exported from those programs.
Just had a thought...you said..
I'm ripping the dvd game films and making them into avi files
So you may be close to the people at the school, know them etc.... Maybe there is a way to get the source material before it is compressed a lot...from the person(s) shooting the games ??
If you talk to them about your difficulty and ask , maybe you can get better source material to edit the stuff.