I am a registered voter in Virginia and I am against this legislation.  Modern facilities are set up with individual privacy in mind and there is no particular reason to prevent people from using whatever restroom is most comfortable for them based on their apparent gender.  I can tell you that it is pretty stupid uncomfortable for everyone if a person who identifies and dresses as a woman to use the mens restroom.  It is more dangerous and more obtrusive than if they just go and do their business and leave.  Fear mongering aside, there is no evidence that trans persons are a danger based on bathroom usage.

As far as sports go if they were set up in a way to be particularly fair to begin with I would have a lot more sympathy for people complaining about trans athletes having an "unfair" advantage.  However, sports are set up in an extremely arbitrary way based on things like "where you happen to live" and "what college you can afford to go to" which isn't particularly fair for anyone.  This is even before you consider the money available and time available to parents who are from different areas.  The most fair systems are probably in wrestling, swimming, and boxing, where teams are often already co-ed these days because the system is set up in such a way that gender in particular doesn't impact fairness.  I don't know that these systems would work for other sports but it would make more sense and be more inclusive of other people (not just trans people) if sports were organized by something besides gender which is difficult to define and socially problematic.  For these reasons, and many others I am against prohibiting trans persons from participation in sporting events (or anything else) based on their chosen gender.