The overall answer is yes, indeed, hypothesis testing is a highly formalised representation that does not reflect reality. Some schools of statistical thought simply do not incorporate hypothesis testing into their paradigm.
Question 1: no, statistics precisely deal with the fact that the data is the realisation of a random phenomenon: in both the Fisher-Neyman-Pearson and Bayesian perspectives, the uncertainty about the data (random) and the answer (not definite) is taken into account. Either as Type I/Type II errors, or as a probability value. In those approaches, the statement is never that $H_0$ is true/false, but that the data "significantly" agrees/disagrees with $H_0$. Obviously, users of such tests may go a long way in misinterpreting the outcome of the test, but I see no point in debating this.
Question 2: a double negative does not help but, no, the concept of Type II errors is precisely including the fact that $H_0$ could be false, in order to evaluate the impact of a wrong decision under both circumstances. This has nothing to do with a "reflection of reality", this is coherent within the statistical model (and the Neyman-Pearson paradigm) but it states nothing about misrepresentations of reality/misspecified models.
Question 3: no and yes. A statistical hypothesis $H_0$ is a question about a probabilistic model observed through an observation from this model (or an alternative one). It is not a question about data in the sense it does require one or two models. However, it is not a question about the real world in that it operates only within the framework of this single or those two model(s).