"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