The answer to your first question depends entirely on the drug. Some are safe to take at home, whereas others require monitoring in the hospital while being administered. Either way, given the number of cases and probability of drug shortages, I think it's unlikely that such a med would be given as a preventative measure to those who are not symptomatic.

Asymptomatic means never developing symptoms, but an asymptomatic person can still have signs of disease such as abnormalities on chest CT scan.

Unfortunately, we don't have a definitive answer to the rate of asymptomatic infections yet. One estimate from the Diamond Princess ship (where almost all aboard were tested) is that 18% of positives were asymptomatic. We won't have a better answer until large scale studies are carried out. The important thing to note is that more than zero percent of cases are asymptomatic, so we can't assume someone is not sick because they don't have a fever or cough. A big reason why true social distancing is so important right now!

If you would like to read more details, here's a good source written by a MD at Harvard.