I wonder if the squeak is some sort of capacitance discharge.
So I still have three unidentified chips... two seemed to be paired together in the box, so I assume they're a pair of buffers? If I had to guess, they might be the LHM6643 since it appears I have the other ones from the opamp kit.
If they're not, do I risk permanent damage if I give them a try?
The pops and other artifacts that you hear when turning the amp on and off are normal, and will vary somewhat with different combinations of opamps. The sound is due to the power supply caps charging up and discharging. Once the opamp has a stable supply voltage, you should hear nothing. Sometimes you will hear a scratching sound as the volume knob is rotated (without an input signal). This is usually due to a bit of DC offset voltage, but, again, this is normal and will vary depending on choice of opamp.
You can safely use any opamp in the buffer sockets, but just make sure that they are both the same, otherwise you will likely hear a different SQ in the left and right channels.
You might consider using the dummy opamps that have just a very tiny SMD resistor soldered on the adapters in the buffer sockets. Many find that this will yield a more natural and lifelike sound while extending battery life.
The LMH6643 buffers will be a warmer, more mellow sound.
IMO, the AD8616 is preferable to the AD8656, whether in buffer sockets or in LR sockets. The highs are smoother and there is little, if any, sibilance.
A good combination that has superb battery life is an AD8616 in the LR socket and 2 dummies in the buffer socket. SQ ranks right up there close to the latest Topkit. ( TLE2141 Class A module + EL8201 buffers).