it is very possible that Legolas in LotR and Legolas in the Sil are the same person.
I'm afraid this is not really possible. Legolas in FoG is an Elf of Gondolin, probably Noldorin (though possibly Sindarin). Legolas in LotR is the son of Thranduil, son of Oropher; Oropher is said to have been a Sindarin prince who came east to settle among the Silvan Elves. It is strongly suggested in UT that Legolas was not yet born when Oropher moved east.