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Yes it's Nandina.

The little shoots coming up around the base of the nandina are normal. These plants spread by runners as well as by seed, so these are either growing from dropped seeds or coming up from the roots of the older plant. You can remove them ... if you remove the seedlings carefully enough, you may pot them up in a little growing medium and give them away if you have a friend who would like them. Just be warned they do spread a bit more than some people would like (and don't forget bird dispersal of the seeds).

Next time you feel you need/want to reduce the size of the shrub, or just tame it a bit, a renewal pruning, rather than cutting back by 1/3, will give best results. With pruning shears or loppers, cut to the ground up to 1/3 of the oldest (which will most likely also be the tallest) canes. Another method is outlined in the link below.

You will probably appreciate this shrub much more when it is covered with bright red berries through the winter.

Here is a link that might be useful: Pruning Nandina