I think the teacher would generally just include their first names as part of the address, e.g.
(But sharing the same family name isn't just a problem for twins. If you're called 木村, 田中, 山田, 佐藤, 斎藤, or similar, chances that someone in your class shares your family name are higher than usual. My guess would be that two people in the same class with the same family name are more likely to be unrelated than likely to be twins...)
That said, some teachers may also use first names and/or nicknames to address their students, depending on the age of the students (and usually also depending on the age of the teacher).
As for your second question, the first born twin is treated as older sibling and the second born twin as the younger sibling. This is also recorded as such in the Japanese family register (戸籍). There is no common name for "[younger or older] brother", only for "younger brother" or "older brother", so you would talk about your twin brother as 双子の弟 or 双子のお兄ちゃん, etc., or even simply as 弟 or お兄さん, etc.
For all practical purposes, there is little difference between twins and (non-twin) brothers, except the fact that they share the same age and thus are in the same class (although in bigger schools twins may also be separated into different classes). In some families this will also mean that the first born twin, if also the first born son, will be the heir.