One way to determine weight is to use the wraps per inch (wpi) method. Wrap the yarn around a ruler and measure how many strands per inch. Don't pull it tight and don't leave it too loose. The table at the bottom of this page
will give you an idea how to classify the weight. This doesn't work well with thin fuzzy yarn which is often knit at a much looser gauge than the thinner core because the fuzz fills in the stitches, but for a loose yarn it's pretty close.
For those tiny balls, they can be sorted by color, or colors that go well together and knotted together to knit all in one item, like a scaf. Let the ends show to make a kind of novelty yarn. Or you can do a scarf, one to 3 rows with each color and leave the ends free for a fringe. They could also be good for amigurumi - little stuffed animals.