If it's sticky, it's not likely stain. Stain just soaks in and dries. Even if the guy slathered it on thick.
I'd guess(almost bet) the guy put some raw linseed oil (vs the boiled kind) on too thickly. It tends not to dry well, if at all. Epoxy bedding compounds will not stick to it.
A rag soaked in turpentine(Wear rubber/latex gloves. Keeps the stink off your hands.) and applied to the stock until the linseed oil starts to dissolve, then left for 20 minutes or so, then rinse with warm soapy water should remove most of it. Repeat as required.
Sadly, you're only other option to remove it is sanding. I'd scrape as much off as possible with a hard plastic scraper first.
Once the stock is clean of the goop, you'll want to seal the inside of the barrel channel. It does not get bedding material past the chamber area. Any wood sealer will do. It's just to keep moisture out so the barrel doesn't get pushed around in different outside moisture levels by the stock warping.
Lose the cotton too. Sounds like the twit tried to float the barrel without knowing how. Floating a barrel may or may not improve accuracy anyway, but fixing that part isn't a big deal. Worry about it after the stock is cleaned. Putting a pressure point in isn't a big deal either. That's just a bit of bedding compound put about an inch or so aft of the end of the forestock with release agent on the barrel.
You'll want this too.
http://www.sako.fi/pdf/manuals/Finnbear.pdf