Generally speaking? No, probably not.
Mostly because AP characters value levels over items in most cases, and in bot lane you'd be having a support sucking up XP. Overall, this would mean that the enemy's AP carry in mid is becoming stronger than the AP character in bottom lane, while the allied AD carry in mid isn't becoming notably stronger than the enemy AD carry at bot. AD carries scale with items, and not really with levels. Thus, the AD carry wouldn't really gain anything from being in mid lane.
The majority of AD carries also couldn't lane against the majority of AP carries. While AD carries would start off strong, with decent pokes, once the AP carry gets to level 6 and has a Doran's Ring or two it is pretty much game over unless they've fed. There is no way an AD carry can keep up with the raw control and damage of an AP mid. Regardless of who the mid is.
The only exceptions to the above rule would be Corki, Ezreal, and possibly Urgot. Corki actually does scale with levels, as does Ezreal to an extent. Urgot is just innately tanky with super long ranged harass. ALL of those characters would have mana issues though, and would have to rush a mana regen item to compensate for that. And Corki and Ezreal would still have trouble with high burst/high CC champions such as Cassio/Annie/Ahri/Karthus/Just About Every Popular AP Mid Right Now.
Specifically speaking? Sure!
There will always be specific situations where you could counter certain champion picks by using more elaborate strategies. This is where a lot of the original "meta breaking" strategies come from. Such as running kill lanes in bot, double AP carries, double AD carries, or even double jungle. It can work in certain situations, but it won't work on a regular basis. And will rarely if ever work in solo or duo ques.
The short is yes. Manamune does include its own mana in it's AD increasing passive.
It should be noted however, that until you have about 2000 total mana the AD given from Manamune is not worth the cost of creating it.
So for your specific question about Urgot I would recommend grabbing a tear early game, but holding off on upgrading it until you hit 1800-2000 mana. In the interim you can get other items such as a brutalizer or an IE if you are doing really well. But like with all things in LoL your build should be fluid to match the flow of the game.
Best Answer
No, you still have to buy the skin itself.
However, if you get the chroma, you can take as long as you want to purchase the base skin. For example, if you received the Garen chroma today, you could wait until next year to unlock the base skin and the chroma would automatically be unlocked at the same time.