Yes, that is a nest. If you're thinking of breeding then I suggest you do a lot of research. There is more to it than throwing a male and female together and letting natire take it's course. Taking care of fry is a big responsibility and time consuming with feeding, cleaning and water changes.
No, most bettas will almost always make a nest when females are introduced, through a jar or some kind of barrier.
I would strongly not recommend breeding, at least yet as it is tedious. Once you are familiar with bettas, breeding, know about genetics, have fry food (they eat live food so if you can't feed live food you really can't breed as they rarely go after other foods). I don't breed because of space and time, everyone has one thing that keeps them from breeding at the moment.