Paul: I thought you were using an acronym of some sort, didn't see that TOPCOW was original posters handle. I just had a little brain fade (kind of like brake fade, but doesn't smell as bad).
What did you buy at auction, a whole car or a motor with a jumped chain? If your parents car jumped at start-up, hopefully it is a simple chain and gear update. I have done a couple that were OK, but unless the owner is willing to pay for a motor tear-down, there is no guarantee that bendable bits didn't get bent. Ya pays yer money and ya takes yer chances.
On the rough idle, that doesn't make alot of sense in a few different ways. Mainly, the idle control module controls how fast the motor idles by adusting the frequency of the idle valve. It doesn't really have much to do with how smoothly the engine runs. By disconnecting the O2 sensor I believe the Lambda would go to 50% fixed operating mode. Maybe this is running the mixture closer to where it runs smooth, but not where it should be for emissions and so forth. I would begin to suspect something amiss in mixture preparation, not sure if it's driving rich or lean, common suspects would be a vacuum leak, bad injector or fuel distributor, or air flow sensor. Best bet would be to get it to a shop so they can inspect the Lambda control with the O2 sensor hooked up and see if it's driving rich or lean and go from there.