You don't NEED it, but depending on your tolerance level, it may make it more comfortable.
Padding would be worn either under OR over chausses. Some of the crusaders mention the upper padding being over top of the chausses. Sir Nathan could give more details, I'm sure. Not really my period, but I've picked up a few things from others.
Padding under half greaves (schynbalds) and over top of mail would have been redundant. I don't think they would have, but just a personal thought, no specific sources to back it up with. My schynbalds have a light skin of leather behind them, but I did that so I had the option of wearing them without chausses, and possibly customizing a set of chausses by chopping out the bits that the schynbalds would cover.