Pictures lose quality and audio in a video clip starts to clip after render

Question

I enjoy doing various random small projects with final cut studio, but there are a few problems I am still having in doing so. One is I record a local basketball league's games and make highlight videos out of them. Another is I like to convert dvds into files and make favorite quote montages or edit movies to have clean versions for the kids. I do not have professional equipment, yet. Final Cut was the first step I chose to take to upgrade. So I'm sure most of the stuff I use is not the greatest choice, but I'm just trying to get the best out of what I have. And when I have problems that I know are bad, I need to find out how to fix them.
Problem with basketball highlights:
I am not sure which is the best format to convert the video to in Final Cut that would make it work the best. Also a problem importing pictures.
The camera I use to record is a Sony Handycam Exmor HDR-XR100. I am confused about importing in Final Cut so I usually import through imovie, which saves the files as .mov and then I go to the finder and put them in Final Cut. Clicking on more info shows me a 20 minute clip (of basketball in this case) is 14.29gb, quicktime movie, 1440 X 1080 dimensions, apple intermediate codec, integer (little endian), timecode, channel count: 2, total bit rate: 102, 385. When I record with the camera I usually turn off the button 'easy mode', which shows on the screen a better quality image going from HD to HDHQ and 5.1 surround sound. Sometimes the render times are ridiculously long for small files and the quality decreases or what not so I was just wondering if there is a good extension to use or format that would be better for use in final cut. Obviously some extensions don't work in final cut, but the most familiar I've used are .mov and .avi (though I don't use avi anymore because I've been told it's not the greatest idea)
Another recent problem I'm having is with pictures in these highlight videos. When I record basketball, I like to record and show everyone their stats along with the video at the end of each game. So I create an excel document with all the teams and numbers and then save it as a .pdf file or picture to load into final cut and add to the video at the end of each game highlights. A similar thing happens though, when I load it in the browser from there the quality is great in the viewer. As soon as I load it in the timeline, the quality drops drastically and I can't even read the numbers in the viewer or canvas. When I took a final cut training class I remember hearing something about a button that makes final cut work faster by reducing the quality of the media it's displaying, but doesn't affect the output. I was wondering if maybe there is a simple button that I have set wrong and maybe that is the reason this is happening, or maybe I just am making it a bad choice of extension again I don't know.
Problem with video files not recorded by me:
Other times I don't record from my own camera say if I want to make a movie montage of all my favorite movie lines/quotes. I would like to edit movies as well to make them clean for my kids to watch. In any case, I convert the dvd to a file on my computer, usually multiple extensions and just use the one that works and also has the best quality, and then load it to final cut. (Is there a program in Final Cut Studio that would be better for this? Right now I use iSkysoft Media Converter to convert these files). As soon as I load it to the timeline and render it, the audio skips like a hiccup throughout the movie. This is when I notice the skipping is only really audible in the audio when music starts to play. I thought that was kind of weird. I can't pick it up as much when there is just dialogue, but when a song comes on or a scene begins with music it makes the entire audio sound bad.
So maybe the problem isn't what extension the file is and I need to look at more details such as dimensions I'm guessing. If you have any examples I would appreciate it. I am sorry I am so difficult haha.
Thanks again for all you're input and help. What goes around comes around and I really try to do the best I can in everything I undertake. I really appreciate it.

Answer 1

I called it "making a montage of my favorite movie clips." It is what it is. Just how I said it. I do stand up impressions part time and sometimes, believe it or not i get recorded, or I record myself and "make a montage of all my favorite movie clips" or scenes from a movie or anything for that matter. So as illegal as that is I apologize that you want to monitor that stuff.
So lets step out of our assumptive sheriff shoes and stop hazardly guessing at anything that you would disprove of. Everyone else has been very helpful. Besides, all these questions centralize around a basketball highlight video I am making for a coach, that I myself recorded and own the copyright to, to be specific and hopefully clarify your hunch.
As a video editor you also often receive footage with numerous formats depending on who you're working with. Not everything is an illegal transaction that you assume it to be. Thanks