1. more theory class. more method class.
For PhD Finance, course in microeconomics theory, time series econometrics, may be a couple of advanced statistics class (in your master if there are any) will be very helpful.
and also take the theoretical and empirical asset pricing class (of course I mean the PhD level on in the business school) if possible.
2. well, finance, or to be said as financial economics, is just a stream of economics. people with a pure economics background at undergraduate shouldn't find any problem in doing financial economics at graduate level if they are in the correct master and correct department.
of course things are slightly different if one is talking about corporate finance or financial management, which is more on the accounting & finance side.
3. I think the answer is quite clear, right?
P.S. it seems that there is an PhD track in the MS right?
go for it with some of the suggested math / stat course.