Question: Are Violet and Mike stuck as being blue and stretched out permanently?
Answer: In the original movie with Gene Wilder, Wonka assures Charlie that "they will be returned to their nasty selves." In this version they are stuck like that forever as a lesson to other children.
Answer: It's unknown if they'll stay that way for the rest of their lives or if they'll eventually return to normal.
Question: If the Buckets were very poor, then why do they even have TV? Why don't they use the extra money for food?
Answer: TV is kind of a necessity. Here in the UK debt collectors can't take a TV if it's the only one in the house. Also, they can't afford newspapers so how would they catch up on news? The TV will last for a while whereas it would only buy a few days worth of food, so the TV is a more sustainable option.
Answer: In addition to the other answer about TV basically being a necessity, there's also the very real chance that they got the TV for dirt-cheap or even free somewhere. It looks like a crummy old antenna TV anyways. They often get old TV's in things like thrift stores or flea markets, etc. I actually specifically look for things like old TV's and VCR's in thrift store and find them a lot since I enjoy collecting old analog media and devices. Plus, given that the grandfather used to work for Wonka, it's entirely possible that he bought it in the past before the family became so poor.
Answer: Don't they have to pay for a monthly subscription? Isn't cable paid for every month?
This was based on the book, not the timeframe of the movie. There was no cable for TVs when the book was written OR in the 1971 original film. Yes, cable was available in 2005 at this film's time but that's not the logic here.
Question: I understand that the video game Mike Teevee was playing was made specifically for the movie. I also know that it's a First Person Shooter game. Is the game similar to any actual real life video games?
Answer: Other than being a first person shooter, no it is not similar to any other game.
Answer: Being an avid gamer, I'd say Unreal 1 is the closest game to the one represented in the movie.
Answer: Looks like a doom 3 clone.
Question: Why would Willy Wonka rather not talk about the pink sheep-shearing room?
Answer: It's probably to be assumed that what they're shearing is what Wonka cotton candy is made from, and Wonka didn't want to state this to anyone else. I can imagine 8 people from around the world that lost the competition going out and telling people cotton candy is made from pink sheep's wool wouldn't sit well with the masses.
Answer: Because is probably just one of those weird and wonderful things, like much of Wonka's world, that he could not really explain simply, thus feeling it was best left unsaid altogether. It may well be a nod to another Tim Burton/Johnny Depp movie where he played Ed Wood, a transvestite who liked pink angora sweaters.
Answer: I saw something saying that the shears may have reminded him of the silver hair.
Question: Does Violet stay blue and Mike remain incredibly thin and tall after they leave the factory?
Answer: There is a scene at the end of the film showing the children leaving the factory. It shows the changes ARE permanent. Mike stays stretched and Violet stays blue. She is also impossibly flexible: https://www.youtube.com/watch?v=jBOvtKI73yc.
Answer: In the book, it's implied that they did.
Question: After young Willy sees his childhood home disappear, where did he live and grow up?
Answer: We are never shown what happened to him but he may have gone to live with relatives or he was simply put into an orphanage.
Answer: It's left vague in the film, but in the book, it's made clear that yes, they are stuck that way.