2. There will likely be some disagreement here, but "either" is technically singular and should take a singular verb. That said, it is not uncommon to hear/read constructions like yours with a plural verb. In formal English, the singular would be preferred by most.
3,4. There is really no difference between "everyone" and "every one". There are differences of opinion about using plural pronouns for singular genderless pronouns. Traditionalists still frown on that practice even though it has become very common. I try to avoid it in almost all cases. Here are some strategies to avoid the practice.
a. Use the singular male pronoun when the gender of the individual(s) is unknown [brought his pen]
b. Use an article instead of a pronoun. [brought a pen]
c. Use both pronouns [brought his or her pen, his/her pen]
d. Restructure the sentence in the plural. [All the students brought their pens.]
5. Again, "neither" is technically singular.