Should I be concerned about the hum when TT is not connected, or can I safely ignore it since I don't get hum under a normal operation?
It may be important or not. The hum without TT connected is an indication of some input shielding problems. You have the power supply with a lot of wires with AC in the same case. In my opinion - it's better to have either very short input wires (for example by rotating the PCB) or shielded ones. To eliminate the input pollution in the further tests you can solder a wire in place of R16 and R34 and do not connect the TT at all until you spot the source of the static noise.
Before going further, please check for static after the wires are in place. If possible, just in case - use batteries for the period of checking.
... I turn off the Bugle PS, pulled the power code out of the case, and connected two 9V batteries to Bugle. I still hear the static. I guess the PS should be fine.
I think - yes, you are correct.
... removed opamps form DIP sockets.
- Removed only U1 - The static is reduced slightly, but not much difference.
- Removed only U2 - The static is dramatically reduced. I can hear only a little of it when I turn the volume to the max.
- Removed only U3 - I don't hear anything, even with the volume at the max.
Does this mean that the second stage is the culprit? Should I inspect the parts and joints starting from R28 and up to C4 on the right channel and the equivalent on the left channel?
Practically - yes. But the order of the checks in my view should be the following:
- use Speedskater's idea of presence a DC offset somewhere in the circuit: the first and the second stages are DC coupled, so it is possible that you have some DC after the first stage affecting the second stage. So - check the DC voltages on the pin 1 (to ground) and pin 7 (to ground). They should be zero or very close to zero.
- you have the problem in both channels. It is likely that the source should be in circuits which are shared by both channels. For the second stage it is power supply circuit (pin 8 and 4). You should check this first (cold joints, bad contact in the DIP socket, C8, C11).
- as far as you don't have the noise after removing U2 the most probable place for the problem (if the previous two checks dont' reveal anything suspicious) - R28, R32, R29, C6 (and corresponding parts in another channel). The second place - R21, R22 (R3, R4). I'd first try to (U1 removed, U2, U3 in place) short-circuit the point where R10, R11, R14 connected together to the ground and listen to changes.
I will try short-circuiting R17 and R35 today or maybe tomorrow. So is what this means to replace those registers with just wires? Are these registers essentially not necessary?
The resistors are nessesary! The easier way of short-circuiting the resistors has been written above.
By the way - very good looking build!

P.S. Speedskater wanted to know if your amplifier has caps on the input (AC coupling) or not (DC coupling).