can you keep a raw egg out for years?!

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As long as the egg stays intact, it won't stink. However, crack that puppy open and it will provide an unholy stench you won't soon forget. At which point there's no Febreze on Earth that will help you. The way you can tell a fresh egg from a rotten one without cracking it open is to shake it near your ear. If you hear liquid sloshing around, it is rotten. If you hear nothing, it is still good to eat.

I don't think it would like room temperature indefinitely. They are (slightly) porous you know. If it were kept in a cool and dark place, it might just dehydrate without much stink, but I wouldn't bet the farm on it.
I have heard that honey is one of those types of food that will keep indefinitely, some (usable?) having been found in Pharaoh's tombs. This may only be folklore. I dunno for sure.

Most likely. It is only when it is broken that you would smell the stench.

I don't think so, I'm pretty sure flies would also dine on it.