As others have stated, you pack them in a few inches at a time so the rope will play out smoothly and without tying itself in knots.
It is good practice to let your rope dry out after a trip so it doesn't mildew or deteriorate. A dry, but shady spot is best (limit the UV exposure if you leave it out for a week at a time).
Second throws are always a little more tricky, but with a little practice, you will find something that works. I like making two coils and throwing half the rope.
If you put water in it, you don't need much, a cup or two is probably plenty. If you hit someone with a half gallon bag of water (or even worse, a bag of gravel) it could hurt or knock them out.
On the river, try to pack you rope while standing in an eddy or a place that doesn't drag your rope through a bunch of dirt and sand.
A rescuer letting go of a rope should be a pretty rare event, so the knot or loop in the end is a fairly small risk of causing a problem. Can it snag, sure, but it can probably be recovered too even if it does.