I don't think it works like that.. if you open a claim while you're within warranty, and the claim/ticket is still open when your warranty expires, they don't just drop your claim. I could be wrong, but I really hope I'm not because that's really f'd up if dealers/manufacturers are allowed to do that.
Diagnosing elsewhere seems like a good idea, but it depends on the dealership too. Some may be adamant about diagnosing the problem themselves because it puts money in their pockets. Some may also require that diagnosis be done by the dealership for a warranty issue because they don't want to be on the hook if the part/repair doesn't fix the problem.
Any dealer (or reputable independent) provides a 12 months / 12,000 mile guarantee for out-of-warranty work. It would stand to reason that a repair done 1,000 miles prior to manufacturer warranty expiration that failed after the warranty expired and within 11,000 miles would be handled under a manufacturer warranty claim.
Now, circumstances may differ. If it was a failure of a replaced part I would expect such consideration, escalating to corporate if necessary. It gets trickier if a problem is reported, say an oil leak, work is done under warranty, and then it is discovered the leak originated elsewhere after the warranty expired. Did they miss it the first time? Or is it a new, different leak? The waters then become muddied, and even more muddy the longer the time between the two services.
As for your second point, you don't need my many years of experience in troubleshooting application software, often times arbitrating disagreements over whether a purported "flaw" is by design or in fact a programming error, to know the first step is recreating the problem. No dealer tech worth his salt is going to take the word of some outside party though that opinion might give him an idea of where to look first if plausible. He must recreate it for himself. For the case at hand, a squeak, that independent opinion isn't going to add anything if the owner pinpoints it himself and gets the tech to recreated it.
When by wife reports problems around the house (electronics as much as anything) I sometimes must report that I cannot recreate it. Long ago I realized that the next statement, "you must have done something wrong" is ill-advised.
Now, it's "show me what you did exactly" which is taken somewhat more kindly. Sometimes some previously unidentified piece of information then comes to light and the problem is recreated.