In this example? Because we know that one is a true positive. We tested 100 people with a false positive, and 1 with a true positive.

If we test again, and the test is "fair", then of these 101 people, we will retest 99% (or 99) true negative, 1% (or 1) false positive, and 1 true positive. So two positives, one false positive and one true positive. 1 out of 2 is 50%.

(Tho, there is a chance also that the true positive might come back as a false negative, then 1 out of 1 or 100% of positive results would be false positive)

The second population contains 100 people without the disease. Of them we expect one person to have a false positive. In addition the person who actually has the disease will test positive. So on average 2 people will test positive, one of whom has the disease and one of whom doesn't. If you are one of those two people you have a 50% chance of your (positive) diagnosis being false.

There is a 1/10000 (the OP said 0.01%) chance of having the disease. That is, in 10000 people, one person will have the disease. But also, if you test everyone with a 99% accurate test, 100 people will test positive. If you test them again, 1 person will get another positive test. So, at the end of the day, your test will identify the one person with the disease with a 99% certainty (or 98.01 if you require two tests to be positive). But also identify another individual as being doubly positive. So, after two rounds of tests, 50% of the marked individuals truly have the disease.

The silly thing with this demonstration is that accuracy is poorly defined. Tests are normally conditional on the actual presence of the disease or not.

Yes, it's not a 50% chance of false positive. It's 50% chance of false, given positive test.