The meaning of a type II error is not dependent on knowledge of the true parameter value, so the direct answer to your first question is "yes". If you accept the null hypothesis when the alternative is true then you've made a type II error (although you won't know it).
However, I suspect that you really want to know about the rate of making type II errors, or the power. The answer to questions about them are more complicated. To calculate the power of an experimental design (and analysis) you have to specify an alternative value for the parameter of interest. Just saying that p differs from 0.5 is not enough because if the true value of p is 0.51 then the experiment will have low power unless the sample is very large, but if the true p is 0.99 then the experiment will have high power with a relatively small sample. The reason that the experimental design aspects relating to type I error rates and type II error rates are so different is that type I errors can only be made when the null is true, and you've specified that condition exactly, whereas type II errors can only occur when the null is false, and you have not specified that condition except as anything except p = 0.5.
Note that the power of the experiment relates to the experimental design, not to the experimental results. Thus you can calculate the power for any parameter value, you don't need knowledge of the true value of the parameter.
The longer answer to your first question (the version that I think you wanted to ask) is that you cannot calculate the power because you have not specified a value of the parameter in the alternative hypothesis. Nonetheless you can still make a type II error.
Your second question about "frequentist repeated experiments" is bigger and my answer may be controversial. The word "frequentist" has different meanings to different people. Some (like myself) find it to be a useful label for statisticians who default to a definition of probability as being a long-run frequency of events. Others prefer to think of frequentists as being those who focus on the frequency of errors of the first type and the second type (i.e. those who work within the Neyman-Pearson error-decision framework). The distinction is important because the first type of frequentist can treat experimental data as evidence that can be used for inductive reasoning, whereas the latter type is constrained to practice inductive behaviour without (or with much less) regard to issues of evidence.
Fisher championed the notion of scientific induction and Neyman championed the notion that experimental results do not allow direct inductive reasoning about the state of the world. (Fisher and Neyman were at war until Fisher died and both played up their disagreements at the expense of their agreements. Much confusion in statistics stems from attempts to amagamate their essentially incopatible approaches.) While they agreed that the meaning of an experimental result related to notional results expected from notional replications of the experiment (your "frequentist repeated experiments"), their conceptions of the type of things that could be considered as repeated experiments differed substantially. Fisher thought of replication of the particular experiment and so interpreted the results of an experiment as evidence relating to the state of the world in that experiment, whereas Neyman preferred to consider replication in a more global manner. To him the repeated experiments were more like all of the experiments than an experimenter might conduct in his or her career. Thus for Neyman the results of a particular experiment are interpreted as being part of a global series of experiments and thus they relate to long term error rates without being evidence about the state of the world pertaining to the particular experiment in question.
