1. You can say 'in the coming days' but it is not often used in everyday English. It's rather dramatic, and more the kind of thing you would come across speeches and in journalism. 2. 'The following week' and 'Next week' do not have the same meaning. They aren't interchangeable. 'Next week' refers to the week after the present one. 'The following week' refers to the week following another specified week. We can use this in the past, for example, 'I went to Rome in the last week of November last year and then I went to Paris the following week' or in the future, 'I'll be in Rome from the 4th to the 8th of April, and then I'll be in Paris the following week'.

"in the coming weeks(years)" and "in the coming days" are perfectly fine, though "days" is much less common. You would usually say "the next few days". The expression is never, ever used with "hours" though.

"In the coming months" is a common way to state an undefined date within the next several months. Using day and/or year would be grammatically correct, but sounds weird to say, (to me anyway :p ) As for your second question, "Next week" is usually used when your explicitly defining the week following the one you are currently in. "The following week" is used when your talking about the a week in the future that is NOT the week after the one your in, but the one after that... or the next "next" week. On the other hand, you could also use "The following week" to describe a week that is even more into the future. For instance, "The first week of next month is when I start training for my new job. The following week is when I will be done." I think of "next week" as future tense and "the following week" as almost SUPER future tense :) I hope my explanation wasn't confusing.... still getting the hang of this teaching thing :P