Analogue (or digital) signal integrity with multiple sources of noise
Can I transmit an analog signal through a noisy environment without it
  being corrupted, or should I use a digital signal instead?


Yes, you can transmit analog signals long distance without them being corrupted, but proper EMI technique needs to be followed. The usual route is to gain the signal up at the sensor and then use shielding on the cable from the sensor to prevent interference from electric fields but not magnetic fields. 10bits of resolution on a 5V signal equates to 5mV, which isn't terribly difficult in most cases but in your environment with AC running close to the cable keeping the signal clean might be tricky.   

The other problem with a cable would be currents (especially non constant/AC currents) on the shield of the cable which must be kept to a minimum or the current through mutual inductance between the shield and the inner conductor(s) can create voltage noise on the inner conductors. There can be problems with running a long cable, such as creating a ground loop. Since you also need to go through a slip ring this could also be a potential source of noise since it would difficult to maintain the shield. The slip ring could be a big source of noise for an analog signal. 

With a digital signal (and differential signaling such as RS485) these problems can be avoided and in my opinion it is easier to isolate a digital signal than an analog one. I would say go the digital route considering your environment. 

If you want to go the analog route, if you had a power supply, you could run an experiment and set it to a known voltage, like 5V and then measure the noise on the other end of the cable with a DMM, if the noise is below 5mV, then it would probably be feasible to go the analog route.