I don't know about the lid. to heat the wax, you can set the globe in a pan of water on the stove and heat it on low heat till it melts(some places I know you could also get a thing called a "candle warmer" which would probably be safe to use because it's made to melt the wax in glass-jar type candles). You want to clean the inside of the globe with boiling water and dishwashing detergent. Then use contact cleaner or similar solvent that paraffin wax is soluble in to clean a second time. You could probably use naphtha based lighter fluid if you can't find contact cleaner. Contact cleaner is usually hexane or heptane based. Naphtha is mostly hexanes as well. Then you wash it with distilled water. You might want to use a bottle brush with the hot water and soap step. I can't ever find one, so I put uncooked rice(it's an awesome abrasive), cold water, and soap in, then shake the hell out of it, then do the boiling water and soap(You can't use rice with boiling water because it would become soft and screw up the whole point of it). I tried doing a similar thing to what you're talking about cutting up the hard wax(I let is harden just enough and squeezed it out of a plastic baggy like toothpaste), but I still got some on the side. When you put it back in, you want to get a funnel with some kind of hose or straw extending it to the bottom(make sure the hose or straw is fastened very secure on the funnel, because if it comes off, wax will go everywhere), then put a couple of inches of your liquid into the empty globe and refrigerate it till it's really cold, then you shake it up so the sides are all coated with the cold liquid, then you melt your wax and pour it through the funnel down into the cold water at the bottom so it solidifies quickly when it goes in. I don't know any of this for sure, I just learned this over the last week from the help of other board members and experience making my first lamp. This is pretty tricky stuff, getting the globe completely clean is probably the hardest part to get right.
I had this problem too. I carefully wrapped aluminum foil around the top half of my globe and left it on the base for a couple hours. It worked for me.