Can Non-termination be considered an algebraic effect?
Non-termination can be considered an algebraic effect up to a point. It's an exception that cannot be handled. More precisely, we may introduce a nullary operation (constant) $\bot$ which signifies non-termination, but then we disallow handling it, as that would allow us to implement the Halting oracle.

Such treatment of non-termination is a bit naive. A more satisfactory account of non-termination would use domain-theoretic semantics, or a lifting monad.