Compressing and adding reverb are two things you can easily do to mix your covers (both of which can be found on Audacity). It won't hurt to play around with different vocal effects as well.
You could also try looking for utaite/youtaite who use the same mixing program as you (Audacity, Mixcraft, Adobe Audition, etc.) and talk to them about it. (I'd be glad to try and help you find someone if you can tell me the program you use.)
As for the original MVs, my best answer would be to contact the original producer and ask if it's okay or check their official websites/blogs. When you contact them, you might as well ask about a download link (there's no harm in asking). If you upload your videos to NicoNico, add the original upload in the Content Tree, and if you upload your videos to YouTube, link it back to the original YT/NND upload. Many producers (EX: Hitoshizuku/Yama) are fine with non-profit use. When they grant you permission to use their PVs, make sure to not only credit the producer, but the artist and video editors as well.
Making/commissioning your own PV is another option.
If you're in need of any tips/answers, let me know and I'll try to help!
If you're looking for more tips/information on starting out as an utaite/youtaite, you can check out these links:
utaitetips.tumblr.com
ytchorus.forumotion.com/t1687-the-beginner-s-guide-to-mixing-audacity