As for stringing chains of processors together, if you read or watch anything about mixers TODAY, you can bet they are chaining digital plugins, and the noise & distortion simply does not add up like with analog gear. Also... those long chains are typically on individual instruments or group buses to achieve a particular effect. It would be rare to put more than about 3-4 things on the final stereo bus.
Although... have you heard modern mixes?? They mostly suuuuck, and the overuse of processing is a culprit. Whether analog or digital, they will (and I will) put several processors on the final stereo bus.
Take for example a typical rock mix.... I might have 50-70 tracks to work with. (not all playing at once though) But 6-10 tracks of drum mics go to the drum bus. Bass Mic & DI go to a bass bus. Maybe 4-8 or more electric guitar parts to Egtr bus. Acoustic guitars to ACG bus. Synths to Synth Bus (could be from 2 to 20 tracks). Lead vocal is likely doubled or with harmonies to LVOX bus. Background vocals to a BGV bus.... Oh yeah... let's not forget an FX return bus for all the delays and reverbs! So I've counted 8 stereo buses in this example.... and each bus has its own processing on it.... typically at least an EQ and a compressor. All these busses then sum into the 2-BUS, which might have, for example, a Tape Emulator, a bus compressor or a mulitband compressor, an EQ, and then a final limiter/enhancer. The order of all these FX also matters.... it matters a LOT! As does the gain-staging... how hard you hit each of these processors. Also don't forget that each individual track can and most likely does have some processing on it. Usually at least an EQ.... sometimes 5 or 6 things. This is why mixing is so much fun!