I know my mum has done it in the past and it recovered well including flowering too though I would scatter a slow release around the roots a month or two before you cut it back to give it a boost.

As this plant seeds around prolifically, I would be potting up some of these as a back up though.

I don't - I just comb all the dead stuff out in spring, but this year I was worried as there seemed to be more dead than live stems. Unfortunately, I then chopped a piece off the side of the very large clump and found out why! There was an ants' nest underneath which had virtually killed the plant - so I've replaced it.