If they really shot down that balloon, and not a Chinese one, they used a 500,000 rocket started by a 200mm plane to kill a $100 balloon. Spending so much on so little on the other hand is impressive.
Usually once every hour or every other hour for a quick second long burst, otherwise they are powered down and just floating. Pico balloons can't carry more than a few grams and depend on solar panels to charge up a battery enough to transmit.
Also air to air missiles are not magic. They don't have secret decoders for every imaginable protocol to lock on to targets (and assume they aren't lying about their location). They are either slaved from the planes radar, contain their own radar, or use heat signatures to lock on.
IE: The AGM 88 "HARM" missile does this. But these are not used by air-superiority fighters like F22, but instead by "Wild Weasel" fighter jets against ground RADAR targets.
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I think the poster you responded to magically assumed that a AGM-88 would work against a balloon. But that is mistaken: the balloon isn't transmitting enough of a RADAR signature for something like HARM to lock into it.
https://dronedj.com/2021/03/24/the-air-forces-training-drone...
Such as - https://www.overlookhorizon.com/wp-content/uploads/2018/09/5...
Detecting them is probably easy--note the balloon material appears shiny. Probably aluminized mylar, it will show up on radar.
As for the heat signature, and I admit I'm doing more guesswork here, it works via IR contrast. As long as the balloon was sufficiently warmer than the air around it, and you were close enough, it should track just fine.
From the missile system and fighter radar perspective, that's not something they heavily (if at all) test these platforms against, because that's not a threat there's any expectation to face.
We might have to actually tax a proportional amount from ultrarich and megacorps, but I'm perfectly fine with the people receiving 70% of all increase in wealth since the 50s paying a significant amount of tax.