Can you please tell me the source of the Minhag that many Minyanim have, to say Lecha Dodi and Kail Adon separately. (either Shliach Tzibur first and then the rest of the Minyan or the other way)?

In the Spharadi communities it is acceptable to verse the prayers. The reason is that the prayer as a Tzibur is one prayer and therefore praying in verses is the prayer of the Tzibur (and not every single has to say the whole song). This, of course is relevant only to parts of the davening where there is no obligation for each person to say the whole prayer, but in Kriyat Shma for instance each person is obliged to say the whole prayer. Rabbi Ro'i Margalit