How did karats come to mean gold purity AND diamond size?
One is Karats (gold) and the other is Carats (diamonds). Legend has it that the carat measurement came about because carob seeds were used to weigh diamonds; they tend to come in a uniform weight (the seeds, not the stones). Merriam-Webster’s does imply they came from the same word:
karat:
carat:
Here’s a little more info: http://www.word-detective.com/back-r.html#carat
This is consistent with the early unit of measure in England being based on the “grain,” since grains (don’t know what kind) are of a generally uniform weight.
Grain was also used as a standard for length, according to Britannica:
During the reign of King Edward II, in the
early 14th century, the inch was defined as “three grains of barley, dry and round,
placed end to end lengthwise.” At various times the inch has also been defined as the
combined lengths of 12 poppyseeds.
If you think about it, this makes sense. The requirement was for a readily available, consistent (relatively) size object which could not be readily modified (someone always wants to cheat). Grain works as well as anything I can think of.
So, a “carat” (with a c) has been standardized at 200 milligrams, eh?
Is a carat a unit of WEIGHT, or a unit of MASS?
Weight. It’s even often called “carat weight”, not just “carats”. You can consider it .07 of an ounce if it makes you feel better, instead of milligrams. 
Oops, .007 of an ounce.
OK, a little more background from OED:
“Carat” as far as a weight has already been discussed above.
“Karat” has the same root, just spelled differently to show which one you’re talking about.
The “karat” came to mean “parts of 24” of gold because it was related to the Roman “siliqua,” which was defined as 1/24 of the gold solidus of Constantine.
That was a specific weight, but over time it came to mean “parts of 24” regardless of weight.
My paraphrase.
Is that .007 of an ounce avoirdupois, or .007 of an ounce troy?
D*mned if I know. I used the very low tech method when I weighed 'em; a miniscule balance scale with tiny tiny counter-weights. If you exhaled the .01 carat (1-point) weights would all fly away; they were remarkably similar to .1 cm scraps of tinfoil.
tracer: I have trouble with ALL ounces. Somehow I’ve figured that an ounce, postal, is some 27-28g. Sometimes I weigh my letters in the lab.
Why, it’s very simple, my dear sunbear: An ounce troy is 480 grains, while an ounce avoirdupois is 437.5 grains.
You DO measure everything in grains, don’t you? 