Just coming in sideways -- I remove (and I mean *REMOVE* in most cases) some pretty weird sounds from already-mixed material regularly. Clicks, pops, squeaks, coughs - even clocking errors (which although arguably the most hard-core sounding, are easiest to make vanish)... It's potentially possible. It's potentially a pain in the butt, but it's potentially possible.
NO DOUBT - If you can remove them on the source track, it's going to be a better option than removing them from the mix.
Totally not trying to drum up business, but I have several clients that forward problematic (individual) tracks for just that sort of treatment so they don't have to roll the dice later. Just saying that the option is there and your tracks might not need to be thrown into the bin.
The difficulty level is completely dependent on the source. On the light side, figure 3-4x real time to go through a cursory removal/reduction. In contrast, I remember one 23-minute project that was billed 23 hours just for restoration (it was a rather high-profile client and the only known source was vinyl) -- One hour per minute of real time. But that's pretty exceptional... And it came out pretty exceptional I might add.
Long story short, there are people out there with the tools and the abilities.