Both basically are the same, although "going to" is more used for events that are more certain to happen in time, whereas "will" expresses and emphasizes merely the intention more. "I'm going to buy a car." "I will buy a car." The first sentence would fit more in a situation where you plan on buying it maybe in the next instance or at a certain day in the week. The second sentence shouldnt be used for that situation but rather to express and emphasize your future, even general intention or will of purchasing a car. Cheers!

There is no difference, both are correct. I think people prefer to say "How are you going to solve this problem" or perhaps even "How are you gonna solve this problem", but grammatically they are both correct and people use both.

Those mean the same thing.