... how important is Damping factor in determining what amp to buy

Important if it is off the rails in one direction or the other but, at normal values, somewhat less important, and....after a point, higher isn’t necessarily better, as others have pointed out.
It is simply a measure of the amplifier’s output impedance, which may be more important with speakers that have impedance variations with frequency, and/or large impedance dips. When you divide the amplifier’s output impedance into the speaker’s input impedance, that is your damping factor. Amp manufacturers often use "8" ohms for the speaker’s input impedance when they calculate their advertised damping factor. This may or may not be accurate, all or some of the time but they have no way of knowing what speakers will be used.
Damping factor is something to be aware of if you are changing out speakers or amps, but not usually a deal-breaker or deal maker.

As the VTL literature explains, achieving higher damping factor/lower output impedance, if done using negative feedback (as is often the case), can come at the cost of reduced sound quality above bass frequencies. A good reason to either bi-amp, or use a sub with a dedicated amp and a separate amp uncompromised by damping factor considerations for the main speakers. That, or chose a speaker not requiring a low output impedance amp to sound as good as it can at both low and higher frequencies.

... and in my on-going search for audio intelligence:
does the age or broken-in-ness of the speaker affect the amps ability to dampen it? (example: right now, my Phillips pre and Onkyo A7 Main-In is powering a vintage pair of JBL L25s...)

If the speakers are not broken in, they are most likely to be over-damped.

My experience has been that is is important, especially for certain speakers. For example, just before the original 801, B&W made a speaker called the DM 16, which had very sloppy bass with a lot of quality equipment, but very tight bass with highly damped amps.
     Consider this. A speaker is an electric motor and can be used as a microphone. On tube and Class A amps, speakers on each end of the circuit can reverse roles as mic or speaker, as on a PA system in my dorm in college. I sent a signal through this speaker into every room in the building, about 300 speakers, from my room speaker.
     BTW, I used the speaker output of a Magnavox suitcase portable stereo's removable speaker, and simply added a female RCA jack to the PA speaker.
       Students were not amused to hear William tell Overtures at 5 AM of the first day of finals.. After my RA told the powers that be that I sometimes played this, I was invited to take a term off, but when I returned, my EE prof seemed impressed that I figured this out.
      My family also built our house, and with some supervision, I had helped wire it, complete with phone taps and speakers for music in every room, speakers that also fed my reel to reel, the tapes of which were quite startling in divorce court.  
       When a signal is sent to a speaker to play a note, the cone vibrates. When the signal stops, the cone is still moving back and forth. and is an electric motor. This movement sends the electrical signal back to the amp, and a highly damped amp will stop the cone sooner.
     Having said that, there does not seem to be a great difference on MOST speakers until the damping factor is below about around 150-500, at least on the Dm 16's. A 150 versus 1000 from the same company, does make a difference, ad these amp use the same pre-drivers and outputs, but more outputs.