Bananas can reproduce sexually (flowers and seeds), and asexually (the pups which are clones, hence genetically identical). The Dwarf Cavendish freely suckers. You may or may not get flowers this season, but this is unrelated to the pups. The mother plant will not die after suckering, and you can either leave them attached or sever them and pot them up separately. When they are mature enough, you can gently tug them apart from the mother plant (they will already have their own roots by this time).

For fruiting, they are ready when they are ready. They want...good sun, heat, WATER, and fertilizing. (Assuming your banana is outside this time of year.)