I believe the stains are actually oxidation and are, in fact, desireable. The oxidation layer "seasons" the pot and prevents off flavors from leeching into the wort as it boils.
I haven't used a better bottle, but it sounds like you don't have enough water in the fermenter. I think it would be safe to boil up some water and cool it then add it until you reach your target OG. However, leaving it alone might be ok as well. It will be slightly higher if ABV, but the flavors will concentrate as well. Take a sample and taste it. If you like it then leave it go. If you think it's too roasty, bitter, harsh, etc. add the cooled water and gently rock the bottle to mix it up. Then take a gravity reading.