Looks like Glossostigma elatinoides.
Yep. Just plug pieces here-and-there and let it do it's thing. There is also Marsilea minuta that's supposed to be shorter. I have some and it's 1" or less and has spread quickly, even growing under shady spots where the ligth is very dim. Either of the Marsilea's or Glosso would make a great foreground.Does it just spread by itself?
You plant small batches here and there, and over time they should merge together and form the carpet ?