Can anyone help me do the above? I can control my set top box with the Sony remote but cannot control the receiver with the cable box remote. I simply would like to control the volume on the receiver with cable box remote, while controlling the channel selector on the cable box.

cable box remote must be agile enough to spit out codes that are in a library that would reside in the remote. Most cable box remotes are not equipped to do that; check owners manual of cable box for clarification on whether your device does that or not.

I'd say no.

The cable box remote uses 3-digit codes. I have found very little info about which codes work for Sony receivers but one post specified a 4-digit code and another specified a 5-digit code.

So, reluctantly, I may be able to do what I need using the Sony remote.

The UR5-8400 only uses 3 digit codes for input on programming. That 3 digit code is the file for the IR database built into the remote.
File codes for other remotes are not going to work on a different remote. Understanding the above should help understand why that is.

There are 7 code sets for Sony audio receivers in the UR5-8400 database (001, 011, 055, 187, 053, 112, 205). I would say the odds are extremely high, one of those code sets will control your Sony audio receiver.
Here is the manual, all the info is there:
[Link: timewarnercable.com]

Prob will adjust volume ok, but don't expect power commands to work.

Tried all seven of those codes. No luck. Selling and replacing the Sony receiver may be the best solution.

Get a nice remote.

The best solution would be a low cost universal remote right?

Get a nice remote- replacing the receiver because the cable box remote (which will probably be upgarded numerouse times with each box upgrade) can't operate it is like selling your car because it needs an oil change..... unless you just want a new receiver and this is a convienet excuse.

I'm going to bet his only source is the cable box. So only needing to control a TV, cable box, and audio receiver - he is having a hard time justifying the purchase of a universal remote.

ASalt,
You can find good deals on eBay, for used remotes that will work beautifully. Most however will require you to learn a bit about programming, to get the most out of it.
I just picked up a used MX-450 for only $40 in mint condition last week on eBay.