Is it a problem (beged isha) for a male to use Rogaine hair treatment to prevent hair loss? Does it matter if this is done for a single person to enhance shidduchim? Finally, if one can use it, is it a problem to apply it on Shabbat?

It is a standard treatment for men and is not a problem of Beged Isha. From what I read on their website I can't see any Halachic problem applying the treatment apart from the issue of pulling out hair, therefore if one can't apply the Rogaine without pulling out hair that may be a reason for refraining from its usage on Shabbat. If you can avoid the treatment by applying right before and straight after Shabbat, that will be better.