Will connecting several mic cables together impact the quality of the recording? Obviously with every additional cable there's an added risk of a loose connection, but assuming my cables and connections are good, is there a reason to avoid this?
While this is basically correct, and I know a few engineers that want to keep cable length as short and clean as possible, in my opinion the quality you lose by connecting two cables together (when good quality XLR cables with balanced signal are used) is of very little significance compared to the type of pre-amplifier you use and the rest of the equipment.
My experience has been that chaining cables together tends to raise the noise level, thereby reducing the signal-to-noise ratio. I try to avoid it whenever I can. It seems to get worse if I'm using an unbalanced cable or a poorly shielded cable. I haven't observed any difference using well-shielded balanced XLR cables.
However, use your ears, and your level meters - try it with just one cable, then try it with a chain of cables. See if you can hear a difference or see that the noise level raises. If you can clearly discern a difference, then you should avoid using such a chain.
A connection adds contact noise and has worse common noise injection characteristics than the cable proper. Also even the best connection has a mismatch to the line impedance of the cable and thus every connection is a point where the signal gets reflected. Now 40kHz has a wavelength of 7.5km in air and probably 80% of that in cable, so you are unlikely to get significant comb filter effects within the recordable (let alone audible) range. So probably the added noise is the main problem and the reflection issue mostly irrelevant.
To get an estimate, try switching your mobile to airplane mode and back (or off and on again) near the cable and either close or far from the next connection. It's unlikely that you won't hear it chattering with the next mobile tower either case, but the volume might well differ.