Zout removed dried blood stains for me and seems to work well on all organic type stains.
I honestly don't think there is one thing that will work for both organic stains and oil-based stains *and* odors. I use a different product for each.
For odors in laundry, this sounds unorthodox but it works really really well:
http://www.amazon.com/gp/product/B000TSO20W/
Yes, it's cat litter additive. I add a heaping teaspoonful to the soap dish in my front loader when I have a load of "smelly" clothes. It has worked every time for me -- especially for that old sweat smell that seems so prevalent in teen clothing (especially athletic clothes). It also worked on dirty clothing left on the floor that the cat decided was a handy litter station.
We don't get many grease stains, so not sure what I would use on those.
Edit: Just read the original post over. I think the cat litter additive would work great on the old people smell, sick people smell, and urine smell as well as stale sweat smell. Might work on skunk, but I haven't needed to try it that way (thank goodness!!!) It might work on mildew and smoke; again, haven't had to try it on those. A container lasts a long time, but you may need to use more per load than I do if your washer uses a lot of water. Mine is a small front loader and a heaping teaspoonful works well with that amount of water.