What are the major sources of loss when sending a signal over cables? What is the impact of digital vs analog, cable material/length and the number and quality of cable connectors? Are there any other major factors that could result in signal loss that impacts audio quality?
There are several major ways a signal can become distorted, these impact both digital and analog signals, but in different ways.
One of the largest sources (on long runs and at connectors) is attenuation, which is the loss of signal strength due to resistance on the line resulting in electricity converting to heat. It generally impacts the signal in a fairly uniform fashion, but reduced signal level doesn't change the noise floor of the equipment reading the signal and thus the SNR (signal to noise ratio, ie the amount of signal that is present on the live vs the amount of noise on the line) is reduced, making the signal either noisier (for analog) or with sufficient noise, more error prone for digital. (Ones may be mistakenly read as 0s.)
Another major source of trouble is induction. A wire of any length passing through a magnetic field is going to act as an antenna (including from the magnetic field produced by the signal itself to the neutral wire). The longer the wire, the more signal can be induced over the length of the wire. Shielding helps reduce electricity being induced in the cable, but doesn't prevent it entirely. Since induced voltages put power on the line in a noisy and random way, they increase the noise level of the line, which also decreases the signal to noise ratio. For an analog signal, this presents as increased hiss on the line, for digital, it can present, if sufficiently bad, as the inability to recognize 0s correctly as the induced voltage may make some 0s look like 1s.
Cables which are made of materials that have lower resistivity will help prevent attenuation. Cables with shielding will help prevent induction.
Connectors can be another source of loss for cable systems. There will always be a certain amount of attenuation caused by jumping from one pin of the connector to the other. They may be in direct contact, but it still is limited surface area and not actually the same material, so there will be resistive loss. This has a much larger impact on analog signals than digital though since that loss won't result in any impact on the 0s and 1s that the digital signal is interpreted as unless there are either a large number of connectors, poor quality connectors, or other problems in the system that push it to the limit of reliably recognizing 0s and 1s.
For digital signals, it is also possible to include checksums in the data stream that will verify it made it from one end to the other without error, but if such systems actually try to correct this error by resending, it will induce variable latency in getting the signal across the line, which could itself potentially be considered a form of quality loss.
One final point of note about analog vs digital systems is that in general, small amounts of signal loss are far more noticeable for analog systems. A digital signal with checksum either makes it or doesn't. Even without the checksum, it will either be mostly intact or almost not intact at all, as the gap between being readable and error prone is generally pretty small.
Analog on the other hand will suffer a quality loss from any change in the SNR, but it will also still have a signal present no matter how much noise you get, it will just become increasingly hard to pick out from the noise.
Every single physical connection will in fact degrade the signal and you will experience at least some amount of signal loss whether its audible or not. By quality I'm assuming you mean integrity of audio from source to medium. In that case the answer is yes.