In an experiment, why can you select your own alpha level for significance testing? From what I have learned, the alpha level tells you that if you repeat the experiment a bunch of times, then (alpha x 100)% of the time you will have a false positive. I'm confused though on if you're the one who gets to pick the alpha level, then how is this always the case? Like if my alpha is 0.05, 5% of the time I will have a false positive (FP), but then if I want to use alpha = 0.5, then 50% of the time I will get a FP. Why is this?

When you do a statistical test (any kind of yes/no test, really), you can have two kinds of errors (wrong results): false positives (test says yes but real answer is no), and false negatives (test says no but real answer is yes).
The catch is that it's very easy to skew (non-technical term) your test one way or the other simply by making the test more likely to give one or the other answer - make a "yes" answer more likely, and then you get less false negatives, but more false positives, or make a "no" answer more likely, and then you get more false negatives, but less false positives.
So, how do you know that your test has a good yes/no balance? There's no "default" or "neutral" setting - there's nothing we can look at and definitively say "this is a fair test". Setting alpha=0.05 is setting your test's skew so that if the real answer is no, your test gives a false positive result 5% of the time. That seems fairly reasonable, so they do it. I don't think there's any reason why 5% is so common, rather than say 2% or 10%, other than gut feelings.
You only get to set one knob - all the other "skew parameters" are determined based on the one you control, and the quality of your data. You can't say you want 1% false positives and also 1% false negatives. If you want very few false positives, and your data is crap, you're going to get very few true positives as well.
Why do we choose to set the percentage of the time we get a false positive when the real answer is no, and not some other statistic, like the percentage of the time we get a false negative when the real answer is yes? Well that just happens to be something that's easy to control.
