It is said that the Earth and solar system are 4.6 billion years old. Presumably this date is achieved from radioactive decay. If this is the case, since most of the radioactive elements would have, to have been in existence prior to this time, how is this date established? I'm sure I am missing the point here, somewhere!!! Is the catastrophic period that the Earth went through, up until 3.9 billion years ago, proved, or just theory, and how was the 3.9 billion date, established?
Do you mean, how do you know the radioactive decay dates the formation of Earth, rather than the formation of those radioactive atoms before they become part of the Earth?
You look at the isotope ratios in rocks, assume that the daughter nucleides were create in situ and work out how many half-lives ago that was. You can't know the original composition of the rock and you can't work out how much of the uranium had already decayed before the rock formed, but if you find gas elements such as Argon/Krypton that are decay products you can be fairly certain that these were all produced by decay and didn't just drift into crystals.
ps. Your second question - the great remelt. When I was a student it was a 'fact' that the earth remelted 3500GY ago due to the decay heat of the initial isotopes. BUT I'm pretty sure I have since seen reports of rocks >3500GY old so either the dates have been changed a bit, the remelt wasn't quite as great or something else. I'm not really in that field so can't answer that.