Kind of, but you're not just picking rationals, you're picking rationals that are known to converge to a real number with some continuous property.

You might be interested in this paper [1] which builds on top of this approach to simulate arbitrarily precise samples from the continuous normal distribution.

[1] https://dl.acm.org/doi/10.1145/2710016

Not really. You can simulate a probability of 1/x by expanding 1/x in binary and flipping a coin repeatedly, once for each digit, until the coin matches the digit (assign heads and tails to 0 and 1 consistently). If the match happened on 1, then it's a positive result, otherwise negative. This only requires arbitrary but finite precision but the probability is exactly equal to 1/x which isn't rational.