I use tap water myself, not really cold or hot, just about room temp.
If the seed you purchased has not been treated you should use a 10% bleach solution and let the seeds stay in the solution for a max of 10 minutes. At the end of that time scoop off the floaters and trash them and rinse the remainder good and let them dry or dry them off some before storing in the fridge until you can plant them. You don't want them to get completely dry, but don't want them to be too wet either.
It's best to plant them as soon as possible after you get them.
The seed coat will turn noticeably lighter (almost white) as it starts to dry out. Add a little water when that happens and stir them up good. I try to stir mine up daily if I have to store them for a few days before planting.
If you have a high percentage of floaters out of your batch (for example 5-10%) of seed that is not a good sign. Could be a diseased batch.
If you have good quality seed you will probably have 20-40 floaters out of a pound (7000 seed). That is what I have seen anyway.
It doesn't really matter. I use a garden hose to fill the container.
Keep in mind, ginseng seed can stand to be much drier than we previously believed and still be completely viable. In most cases, commercially raised seed will be so dry out of the stratification boxes that it will float. For that reason, many soak the seed for several hours or even over night and stir it up good before they discard any seed. Even then, I sometimes crack seed open that floats to be sure it is indeed bad seed. Anything other than firm, creamy white inside the shell means its bad.
In recent years, I've advised against adding water to seed unless it appears to be drying out inside the shells. Dry outside isn't going to hurt the seed's viability. Adding water drastically increases the chance for introduction or spread of disease. But, TN is right, plant it as soon as you can.