Alternative hypothesis

1. Yes, that's a mistake; an equality should not be present in that alternative; it should be strictly an inequality for this. You need null and alternative to specify mutually exclusive sets.

2. There's no requirement in statistics that null hypotheses be point nulls -- composite nulls are certainly a thing; any number of standard books on statistical inference are unambiguous about this (for example, see Casella and Berger, which is pretty much a standard text with a long history of use in real statistics classes).

   You compute type I error rates for the worst case under the null, which is at the boundary (i.e. at the equality) but the phrasing of the hypothesis in words pretty clearly appears to imply a composite null.

   That said, null and alternative also don't have to exhaust the real line (again, see Casella and Berger).

For example, two point hypotheses (null and alternative) are entirely possible and meaningful (see the Neyman-Pearson lemma, a pretty fundamental result in the theory of hypothesis testing). Such situations come up in physics sometimes: for example, just a few days ago I saw an astronomer talking about two hypotheses, one with a parameter at 0 (the current, conventional theory) and another with the parameter at 1 (given suitable choices of units). Non-exhaustive hypotheses are completely okay in the right circumstances.

There's just no suggestion in the question that the null and alternative don't exhaust the possible values for mu, so presumably they should in this case.