Should someone spend EXP they have in a fiction?
Probably not, the Blackburn is a good example of one where that's not appropriate. On the other hand it's easy to see a fiction being started for the purpose of character "X" showing that he has power "Y". It's also possible for some fictions to last months. But those will be the exceptions and not the rule.
Should you gain EXP in a fiction that takes place at the end of a month..If the fiction is still active the next month?
What difference does it make?
The problem isn't gaining exp, it's spending it. The Blackburn took place in late March. On April 1st I got enough experience to buy that item I want. That doesn't let me use it retroactively, and I don't have the ability to spend experience there anyway.
Should characters made after the fiction be let into fictions?
Depends on the fiction. In theory I'd say no for the same reason you can't spend exp... but in practice normally it's better to have more characters than fewer so the usual answer would be "yes".