Li-Ion and IMR don't have a memory.
They do get damaged if discharged too far. That's why they have a cutoff point. It is normally considered best to charge before cutoff. Not pushing it to its bottom limit, so to speak. That said...the cutoff is there for a reason and built into the device for that purpose.
I have historically rotated through 3 sets of 2 14500 batteries (this is for a removable battery PV) and they have lasted well over a year and a half. I use them until they cut out, then charge. Some sets are probably close to 2 years old and still work. I'm going to trash them as performance does degrade with all Li-Ion over use-time. However, it doesn't seem to be a major problem. Probably depends more on the exact device and how fast/slow it charges them...although battery quality varies too.
Charging batteries generates heat. Heat degrades Li-Ion cells. The deeper the cell depletion the longer it has to charge and the more heat (probably oversimplified. Someone will pop in with a link to Battery University in a few posts, I'm sure.).
So...
A) Depends on the device, but most are probably fine if run to cutoff despite a lot of people who constantly check voltage and charge at X.XX volts
B) How worried you are about it depends on the cost of the device. My $5.00 batteries don't freak me out if I have to replace them. OTOH if you're worried, its OK to top it off early. It isn't mandatory though.
