Have you been told by your service department rep that your new car warranty will be void if not changing oil at 3,000-5,000 miles?

A new car warranty is pretty short anyway and your engine should easily not suffer any problems during the warranty period. I have never taken any new car I bought back to a dealer for routine service. They charge too much and you don't have to.

I have been told this and have replied that I want it in writing, signed, dated and will be forwarding it to the synthetic lubricant provider corporate legal department. (The subject never comes up again.)

No - the oil change interval is in the Owners manual - the recommended change intervals are valid just document the oil changes you do yourself.

While a synthetic oil doesn't break down as quickly as conventional oil, it still gets dirty or contaminated just as quickly as a conventional oil.

So if you have a dirty engine, a synthetic oil won't last any longer.

Nope, 20k on my BMW as long life synthetic used as stock, 10K on my VW as normal synthetic used.