I'm 99% sure I heard someone say that Bergeron literally doesn't want the C while Chara is on the team, and I can get that. Multiple, multiple interviews with teammates, former teammates, other captains, coaches, and GMs all point to Chara being an excellent leader. Without being in the locker room, none of us can outline all the specifics as to what he does, but we can look at the results: 13 seasons, 3 trips to the SCF, 1 win. Especially if you consider the tools he was given in the early years, that's a pretty good record.
So, Bergy getting the C before Chara left would be awkward at best, and detrimental at worst. I also know that Chara has referred to Bergy as his co-captain on more than one occasion; I think I've seen it on Chara's Instagram. There's no need to pass the torch. Bergeron has been co-captaining behind the scenes for years now, and when the time comes, he'll be a great captain.
There's no way to predict after Bergeron. Trades, acquisitions, etc., are far too unpredictable to even hazard a guess. Granted, it sadly may be sooner than later - there are only 3 active NHLers older than 40 according to Wikipedia (and, no, that doesn't include EBUGs). If Chara retired now, that would give Bergeron 5-6 more years before he gets into that stratospheric range.