1. The level of generality depends on the way the self-efficacy questions are framed. If a scale is constructed with items like "I believe that I can successfully complete any task I set myself," it is measuring a very general level of self-efficacy. But if the item read "I believe I can successfully train myself to run a mile in under five minutes," it is much more specific (in this case, to athletics). As a rule, relatively general scales allow for weak predictions to be made of a wide range of behaviors, while highly specific ones allow for stronger prediction within their domain, but none at all outside of that narrow range. (By the way, I don't think anyone has ever empirically differentiated "self-efficacy" from "self-confidence.")
2. Referring to self-efficacy as a trait is a misnomer. Bandura was not working within a trait model, but instead a social-cognitive one similar to Julian Rotter's expectancy-value theory (though Bandura was not one to reference earlier iterations of similar ideas). In this sort of personality model, behaviors are construed as learned, and expectancies (of which self-efficacy is one type) refer to a person's subjective estimate of the probability of success or attainment of reinforcing outcomes. Because we accrue a great deal of experience in certain situations, we generalize from that experience. These generalized expectancies can be quite broad in scope and can acquire a trait-like stability, though they are always open to modification.
There may be some tasks that may need more generalized measure of self efficacy to be sensible in the broader concept of what it tends to measure. For example, when we are talking about walking as a task in the context of physical activity and health, breaking it down to small tasks such as walking in a park or walking to work or walking on a track would not make sense in measuring self efficacy. Even though a person may find oneself incapable of walking on a track (maybe due to absence of one, or other reasons for that matter), one may still complete enough number of steps, strolling down the street to the grocery store. In such case, extremely task specific self efficacy may not be useful, but a rather general self efficacy measure for walking. Even though measuring self efficacy for ' walking on track' may be useful in some context, however general walking self efficacy may be useful in its own context. This may be true for other concepts that may involve working over various domains. In this case, measuring self-efficacy as only a task- specific construct, and limiting its generality might reduce its scope and utilization in various other broader concepts where it could be used.
Self efficacy in generic terms might mean the generic ability of defining a problem, critical capability of reviewing various options in holistic terms and context and ability to choose one of the better options as nearest to ideal in terms of innovation, efficiency and effectiveness. Specific one will require these cross cutting together with problem specific knowledge, skills and competence.