In the sentences you give, the first is correct.
Do any of your students speak English?
This is considered to be plural because it's possible that a large number of the students speak English.
However, you would say:
Does anybody speak English?
Anybody always takes a singular verb. The same goes for nobody, somebody and everybody.
None can be used with a singular or plural verb, depending on what it refers to.
Is there any cheese left? No, there's none left. Cheese is uncountable and takes a singular verb. So does none.
Are there any chocolates left? No, there are none left. Chocolates are countable and take a plural verb.In this instance none also takes a plural verb.
I hope this will help you.