Yes. There are references to the thoracomachus or subarmilis. This was some kind of quilted linen. It was likely paired with some kind of hide covering to prevent water from getting into it. It would be worn underneath metal armour to cushion the impact of the blows (sure the sword didn't cut through your mail shirt but without some kind of padding all that energy will still be transferred to you). Obviously none of these have survived as they're made of perishable materials.

As for the Celts, almost certainly if they were wearing mail shirts (and the Romans got mail from the Celts). Would the poorer rank-and-file wear it? I dunno, hopefully one of the flaired redditors will jump in.