In regards to the copper area for the LM317. How is the length determined for sure?

Datasheet (figure 4) tells, that TO-263 package shows a thermal resistance between junction and ambient of about 35K/W, when soldered on 1 square inch copper area. Means if you solder TO-263 package on a copper area of 2.5cm times 2.5cm, then junction will have a temperature being 35K/W hotter than ambient. And as heat dissipation was calculated to be 0.47W maximum, junction will be 16.5K hotter than ambient.

Unfortunately, you use a coating, which can degrade the cooling capability of this copper plane. As result, the copper can heat up a bit, which results in a temperature rise of coating material. As long as I do not know, what kind of coating material you use, and to which thickness it is applied, I cannot make any estimation. But I would recommend you to use an as large copper area as you can provide.

You can make a test, whether copper area is large enough or not:
Apply overvoltage, so that your current limter will invoke. Now, put your coated board into an environment having maximum ambient temperature as expected in field use. Now let the board burn some hours. Test the temperature at copper area from time to time. Can you touch it for longer time? Then temperature is less than 50°C. If it is hotter, then coating material could eventually degrade. Then, use a larger area.
You could also remove the coating material after this test and check the consistence at the place, where it covered the copper area. If you can notice any change of consistence, then temperature was too high, and copper area must be increased.

Of course, you can also mount some temperature sensor directly at copper area, and make the above test. But these measurements are not very reliable, I experienced.

High temperatures of semiconductors are the major cause of too early failures! So, it's wise to keep their junction temperature as low as possible.

Kai