Teacher was likely the one who killed Rider. Probably has Caster's familiars/clones following him while she works at the temple. Caster probably enchanted him, then he sent her familiar to go occupy (and/or kill) Shiro and Rin so they couldn't uncover his identity. Even in the older FSN series he didn't go about trying to kill them, he actually remained in secret until they uncovered him. Quite possible that he's just that strong in this one.
Honestly, other than the animations, I kind of enjoyed Fate Zero more, the interactions seemed more genuine and it's more of an actual scramble for the grail rather than FSN's "save the children" approach...