I doubt that Godzilla knew the egg was there previously. If I remember correctly, when the humans disturbed the egg they caused the strange plant[s] on it to give off vibrations [I believe the dub refers to it as singing] which caused the egg to react.
Hrmm, how can I put this better...
There are really birds called Cuckoo birds -- unlike most birds, the cuckoos don't hatch their own eggs. Instead, they'll lay eggs in other birds' nests so that the 'victim' bird hatches the young cuckoos. I believe that these 'victim' birds also raise the hatchling cuckoos as their own until they kick the younglings out of the nest. [There's more to it, but you can find that out on you're own]
I believe that Toho may have been trying to apply a similar trait to Godzillasaurs [the pre-irradiated form]. Though in my opinion, they didn't accomplish this very well.
Extrapolating on the concept - it would appear that a pregnant Godzillasaur would sneak and lay their egg[s] in the nests of other creatures so that these 'victim' creatures would be the ones expending energy into hatching the egg. But, going by Godzilla's subsequent behavior after Baby is 'awakened', unlike the cuckoo, Godzillasuars rear their own young. Seemingly, when an infant Godzillasaur reaches a certain point in it's development it calls out to any adult Godzillasaurs which would then seek out the youngling to raise.
I don't like this concept much [as applied to Godzilla]. The behavior pattern seems to belong to that of an opportunistic prey animal, and I don't belive most animals of this type are very big. Godzillasaurs [as seen in GvKG] are large animals, it doesn't seem to me that they'd be very good at sneaking up on other creatures nests. Even if, like elephants, their normal walk is fairly quiet, I imagine that ground vibrations would likely alert expecting parents.
Further, even if my hypothesis about what Toho was attempting is correct, it doesn't seem too likely that land-sea animal would use the nest of a flying animal. The whole scenario would've made more sense if the Godzillasaur egg was placed in the nest of an Anguirus-type, Baragon-type, Gorosaurus, or possibly a Varan-type creature. They're all land type kaiju, except for Varan being a land-air mix and it seems to me that their nests woul've been more accessible than a Pteranadon nest.
And on to your second question. I doubt very much that Rodan was aware that egg wasn't a pteranadon egg.
