Guess I'll toss my couple cents...
Teemo can't carry because he's out ranged and it doesn't help most don't build him carry, they build him bruiser-like with a Runaan's and a FM. These items are nice, but they're not what a carry should be focused on.
Teemo can't AP Carry because his ulti is a shroom that's more a nuisance, though it can be very, very devastating when you go less AS and more AP like getting a Rabadon's after a Nashor's tooth rather than an AS item like Malady. He also only has 1 TRUE ability, making him more of an AA-Based AP Carry like Kayle. Your Q does damage, your R is utility, and your E is your bonus AP damage and W is to GTFO or to get in there quickly. What's different is his E actually has a smaller AP ratio and does not splash, which is why I disagree with the previous nerf to make it go from .4 to .3.
He can't Bruiser because he's squishy as all hell. Let's face it... The poor guy has such naturally low defenses that building him tanky mades him do no damage, but he'll always take more damage than anyone else.
He caaaaaaaan jungle. I've done this multiple times and in all honesty, I actually got the idea after watching the troll spotlight. Well, you try to avoid mirroring your opponent so you take a buff and use ghost to get to their's. You'll be level two and you can sit near their blue or red invisible and wait for the opponent to get it low so you can instantly smite it and focus fire on their jungle. He's a great counter jungle, but that's about all he can truly do, sadly so his jungling CAN and WILL fall off unless your team can win their lanes since Teemo is all about keeping their jungle out of the lanes by pissing him off and making him want to rage quit.
He can't support WELL. He can guarantee FB, unless your team pulls an idiot move, if their team doesn't get a pink. Do it the same way you'd do it mid. With the lack of a hard CC like Sona and Ali, a lack of tankiness to allow him to poke like Nunu, lack of an aura or a heal like Nunu, Sona, and Sora... Most importantly, Teemo's downfall as a support is that he can't AoE hit everyone or give a quick buff to an ally so he can get that assist that all supports feed off of.
That's about all I can give off of my adventures of doing Teemo. Jack of all trades indeed, and only a master of being bait. Damn is he good bait.