Have you tried Davinci Resolve? It's a fully featured Non-Linear editor, with color grading and full audio mixer and best of all for the non studio version it's completely free
(There is a paid version but that includes 4k output, noise reduction, and the Fusion compositor)
It does have a bit of a learning curve, but there are plenty of tutorials out on the web for it. Does need a higher spec graphics card if you are doing intense grading, but for mixing audio to picture should run well on most machines.
I think it beats programs like Premiere and Final Cut Pro hands down and all the more impressive that it's free.
Quote. Add audio files to an MP4 video?
Yes it can do MP4 and many other pro formats.
QuoteAdd several audio files to the same exact time-stamp
IIt can layer up to a 1000 audio tracks.
QuoteHand-pick time-stamps where I would like the audio to be placed?
I haven't gone too deeply into this myself but I think there are many tools for placing and syncing audio.
QuoteA vignetting or clipping option. In other words - to cut-off a top and bottom piece of the video
Yes, it's called Output Blanking
If you tried it and found you wanted the Pro Studio version, go to
http://www.bmcuser.com - users who have bought two or more Blackmagic cameras sell off their extra dongles dirt cheap.
But the free version has probably got more than you'll want