Former C-130 Flight engineer here, it was my job to calculate performance data for takeoff and landing.

Rain only matters because it increases stopping distance. It doesn't affect engine performance, and it never even occurred to me that a wet airplane would be heavier than a dry one. To get a sense of weights we care about, our empty weight is ~90k pounds and our max (peacetime) takeoff weight is 155k pounds. Performance numbers are good for 5000 pounds, so even if the water weighs several hundred pounds we'd never notice.

The most important factor in engine performance is the density of the air, which is driven by temperature and altitude. You get more power on cold days and at low altitude, less on hot days and high altitudes. (Which is why Denver has long runways)

There is a decision tree when planning a takeoff, and extended stopping distances due to a wet runway sometimes pushes you to use a higher power setting which is a bit less efficient. So the answer to your question is maybe a little, sometimes, but not in the way you think.

Total guess but I imagine the increase in relative humidity impacts the combustion efficiency of the engine as well

If you consider the fuselage a cylinder and calculate the surface area of the lateral surface it’s 2pir*h. this site has the length as 209.08ft and the diameter as 20.3. That means the fuselage surface area is about 13300ft^2. That same site lists the wing surface area as 4605ft^2, for a total of 17905 square. Assuming an 1/8” of water accumulates uniformly, which is a bad assumption, that’s 2238 cubic feet of water. Each cubic foot weighs about 62 pounds, so that much water weighs 136000 pounds. The normal takeoff weight of a 777 is 534000 pounds, yeah that is a lot. However, only about half the surface area is exposed to rain and 1/8 inch is a lot. Id imagine it’s less than half that weight.

I don’t know much about it myself, but I would guess it is negligible. Maybe for small propeller machines with a fairly limited amount of fuel capacity; but larger planes, especially commercial ones, have reserve fuel for quite some time.

Situations where landing at the destination is temporarily unavailable, air traffic requires the plane to circle for some time, or they are even rerouted to a different airport can always occur and are accounted for. A minor increase from rainfall shouldn’t make a dent. I would think.