The real criteria for replacement are performance based. Does the bike still shift well in front and does the chai run on the chainring with no skipping? If so, the chainring is fine. I've put over 30,000 miles on Shimano chainrings and they still shifted well and ran quietly so there was no indications they needed to be replaced.

Chainrings are much more wear tolerant then cogs since the chain is applied to them under tension and they have a larger radius and more teeth to share the load.

Your 7800 ring will fit since it has the same bolt circle diameter (130 mm) as any current Shimano standard road crank. I assume you didn't order a compact ring (110 mm BCD) unless your 5600 crank is also a compact.