According to Rav Schwab ztl the tune of “ledovid boruch” sung in Breuer’s on motzai shabbos dates back to the beis hamikdash.
I also heard that the “nah nahs” that many shuls insert in the davening on yomim noraaim (between phrases that the chazan sings) dates back to the geonim.
as far a trup – remember that different kehilos have different tunes – ashkenaz/syrian/morrocan/yeminite/yekke etc…