Must one use a vessel to wash the hands after the above actions?
Some Poskim[1] rule one is not required to wash his hands using a vessel, [and hence one may wash his hands by simply turning on the sink or dipping his hands in a vessel]. Others[2] however rule one is required to use a vessel. Practically the custom is to wash with a vessel after using the bathroom.[3]
[1] Elya Raba 4:12 in name of Makor Chaim, student of Rav Chaim Vital; Soles Belula 4:14; Machazik Bracha 4:6; Ben Ish Chaiy Toldos 16
[2] Shlah Shaar Haosiyos 9; Lev Chaim 1:63; Chesed Lealafim 4
[3] Minchas Yitzchak 5:96; Mishneh Halachos 5:2; Halichos Shlomo 20 footnote 86; Piskeiy Teshuvos 4 footnote 171