How can I repair leaks in metal wrapped wood gutters?
Hardness is a good way to check wood for damage without pulling it apart.  Take a nail and press it into the wood where you know the siding is in good shape.  Then, do the same where you suspect damage.  If the wood is softer, then you can assume it has some amount of water damage.  If you want a more direct, verifiable method your best bet is to yank a shingle or two off and check behind them.  (From the pictures it looks like you have shake but its hard to tell.)

Also, as Steven commented...remove those "gutters" as soon as is humanly possible and replace them with actual gutters.  Each year you wait things will get exponentially worse.  If there is not currently water damage (which I doubt, sorry) there will be very soon.