Q3: what is the engine idle RPM at cold run after initial cold start up?
Q4: does the engine run smoothly at the lower idle speed when cold?
Q5: what is the engine RPM that the increases to after it reaches operating temperature, and does the engine smooth out?
Generally the engine speed is controlled by the throttle linkage. At idle setting, the carburetor throttle plates might be closed, with only a small hole in the plate to provide air for idle speed, often referred to as a choke setting, which creates a richer fuel mixture to aid in starting. When the engine warms, the choke mechanism may intentionally open the throttle plates more than at full-choke, which allows more air into the engine. The result is a air-fuel ratio that is better (not overly rich), and with more air the engine runs faster.
Another possibility is as the engine warms up the throttle linkage becomes more flexible due to getting warmer, and it opens the throttle plates in the carburetors more than when cold; again the result is an increase in engine speed.
It is also possible that the carburetors are not in good adjustment to act in unison (or in "synch") and when the throttle position is slightly increased from removal of the choke, the engine speed increases because all the carburetors are contributing proper fuel amounts, and the engine runs faster and more smoothly, whereas before one carburetor was not properly adjusted and it caused the idle speed to be lower than expected.
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What would cause a four-stroke-power-cycle 50-HP Mercury engine with four carburetors to shift from an idle speed at cold start up to a faster idle speed once the engine temperature rises to operating range?
If you add the exact model number for your engine, the response from GROK.COM may improve.
Also, it is very likely that the model year 2000 engine that you have is built with a Yamaha powerhead, and the carburetors and choke system could be from Yamaha. At that epoch Mercury was not able to manufacturer four-stroke-power-cycle outboard engines other than a 25-HP model, and it bought engine powerheads from Yamaha, then adapted them to a Mercury midsection and gear case, and used different wiring harnesses to mate with Mercury accessories.
The four-stroke-power-cycle engines with four carburetors of this era were somewhat infamous for having problems with the carburetors. The difficulty was generally thought to be related to the very small orifice openings on the carburetor jets, which was then thought to lead to those orificies being blocked by small debris in the fuel. On that basis, you might take a first step to help the carburetors by using some fuel cleaner additive in the gasoline. Unless you are a dedicated do-it-your guy and are familiar with carburetors, I would not start tinkering with them without some skill and knowledge in that art.