Creating Custom Games

Is there a mechanism by which I could create a custom game to be saved or played against someone else with time and/or piece odds?

You can use the board editor lichess.org/editor and once you have edited the board, just press the button that says continue from here, and choose if you want to play the computer or a person. You can do time odds by looking at the right of the chess clock and pressing the blue button that shows a + sign. If you haven't configured that, go here: lichess.org/account/preferences/chess-clock, scroll down until you see something that says "give more time" and press always