I'm preparing classes on the beginning of Bava Metziah and I would like to present the information through the lens of the Rabbonim that used the Gemara to answer Sheilos. Does anyone know of a Sefer that compiles Sheilos U'Tshuvos through that lens?
I don't know if any of these three is exactly what you are looking for, but I'm sure that they would be extremely helpful for your purposes.
A Sefer that often quotes various Teshuvos and their explanations of the Gemara is Daf Al Hadaf (edited by R. David Abraham Mendelbaum, R. Joshua Lefkowitz, and R. Abraham Noah haLevi Klein).
As DoubleAA mentioned above, for Bava Metzia, you have the Frankel Raza Deshabsi, which is an index of a huge amount of Jewish literature, including Shu"t, organized according to the Gemara.
Hebrewbooks.org is available to everyone who is on the web. When you open the 'shas' tab, There a Sefer on the mefarshim list called שדה צופים which quotes many Sheilos Uteshuvos the likes of the Radvaz, Chavos Yair, Rambam and many more. I was recently learning Nazir which has very little Mefarshim and I found a wealth of information from the above which I quoted on the daf, absolutely recommend.
Thanks to wfb, here is the PDF on Bava Metzia.
A popular set of seffarim written explicitly for this purpose is שערים מצויינים בהלכה by Rabbi Shlomo Zalman Brown.
His son Rabbi Chaim Elazar finished off many of the Tractates that his father left in manuscript form. The volume on Bava Metzia was one of those.