For the first one see here: Weighting 12 Pool Balls That puzzle is harder than yours, because YOU know one is lighter.

It also depends if you are using a balance beam, or normal scales that give you direct measurement.

In a nutshell, you could weigh 6 vs 6, then the lighter group has the light coin, then 3 vs 3, then 1 vs 1 (if they are the same then the one not measured would be it)

An LP spins at 33 1/3 rpm (revolutions per minute),  is about 30cm wide, and playing time per side is about 30 min (all this I got from Wikipedia)

Ther is usually just one long spiral groove on a side. The length of the groove will vary, because playing time, groove position, etc will vary. So we can only estimate it.

To estimate the length of groove, imagine thay are lots of circles. How many circles? Well, if it does 33 1/3 revolutions per minute and there are 30 minutes, that makes 1000.

If the average radius is 8 cm (I am not sure of that, just seems reasonable on a 30cm wide LP), then the average circumference would be 2 x π x 8 = 50 cm

So 1,000 x 50 = 50,000 cm, or 500m

(Possibly ...)