I have downloaded copies of those five images (and rotated three of them) for careful study.
Looking at images numbers 4 & 5 I can see that the flexible lead to the telephone and modem only uses the red and green wires. (I am assuming that there is a low-pass filter for the telephone and the filter, telephone and modem are connected correctly at the other end of that flexible cable.) The internal cable, used to connect the rectangular box with the circular box, looks to be standard 4-core, 2-pair cable and, I would it expect it to be equivalent to CW1308 specification cable. Assuming that they are two twisted pairs, the colouring of the wires' sheaths could cause problems. I see a blue wire & a white wire making up one pair and an orange wire & a white wire making up the second pair. It is the first pair (blue wire/white wire) that is being used. If I was being super-critical, I would mention that there should not really be any exposed metallic conductors showing -- look at the ends of the blue & white wires where they are connected under the screw terminals that hold the green & red wires, respectively.
Looking at image number 3 I can clearly see that the incoming service feed cable (with the black sheath) is a 4-core, 2-pair cable. Once again, the wire colours could cause a problem because they appear to be a red wire & and a clear wire as one pair and a blue wire & a clear wire as the second pair. It is the latter pair (blue wire & clear wire) that is being used and it is connected to the blue wire & white wire pair of the internal cable by means of two gel-crimps. The unused pair of the service feed cable (red wire & clear wire) and the unused pair of the internal cable (orange wire & white wire) are left unconnected. Again, if I was being super-critical, I would suggest that the unused orange wire & white wire pair in the internal cable should not have been left with exposed conductors, twisted together.
Looking at image number 2 it seems as if the service feed cable has been threaded through the existing hole which contains the wire for the door bell. There is not a lot that can be done -- unless you are prepared to have those two cables separated and a new hole created for the service feed cable.
Overall there is nothing particularly "bad" about that wiring and I can't see it being a cause for a degradation in the broadband service. I know that you have noticed that errors occur when somebody rings the door bell. One possible solution would be to fit a spark-suppressing capacitor/resistor series pair across the push button.