Think i answered yesterday on your question but that thread got deleted somehow. Anyway, we fly the fail operational variant of the NG (both 700 and 800) since around 2005, CAT IIIb down to no DH/75m RVR and CAT IIIa to 50ft DH and 125m RVR in the first segment, 75m for the rest of the relevant distance.
Required for the CAT IIIb operation is the Rockwell Collins Enhanced Digital Autoflight System which does have a rudder channel. Of course rudder servos are needed to utilise that. The Collins MCP which includes the autothrottle computer and FCC is needed as well, however as far as i know that is standard equipment since a certain line number. And of course the ISFD is needed as a third attitude source since the 737 only has two IRSs. It is still a dual autopilot installation, not a three channel one.
During dual channel operation the rudder will be used by the autoflight system below 1500ft AGL (to be more precise after the successfull confidence test) and will stop working during G/A whenever a new lateral mode is selected, either automatically or manually. Especially during OEI operation better be prepared or there will be a violent yaw when that happens.
We use the 26k variant of the CFM-56BE on the 800s and 22k on the 700. On the 700 we do not have any problems at all and can use any weight up to max take off weight for landing and go-around with quite a margin, on the 800 we have to check performance for everything above 65t. However we do have the short field performance kit installed on our 800s which helps quite a bit.
Interesting to read above that apparently some pilots pay for their work equipment out of their own pockets, wonder when they start asking you to bring your own 737 to work. That is something neither me or my colleagues would be willing to do, either the company needs it than they have to buy it themselves, or they dont, in which case it is their decision if they buy it or not. Quite contrary, if the equipment gets more complex and allows operation into situations that were not possible without it, it is a clear reason for higher pay.