If the X37 is public they probably have something bigger and better tucked away in a hanger somewhere remote. Think about how long the F-117, SR-71, U-2, B-2, or RQ-170 were kept secret. There is no way they haven't thought about how to use this in a military capacity.
If they do, it's almost certainly never been put into orbit. The guys who track satellites as a hobby are good.
For sure. I've read that the RQ-170 was seen quite quickly by plane spotters - likewise with the planes used for extraordinary rendition.
Never underestimate passionate nerds and their ability to notice unusual things!