Does Richard Dawkins Violate the Basic Rules of Logic, Science and Reason?

According to both Google online definition and Merriam-Webster dictionary definition, "God" is primarily defined in the modern age as Creator of the universe.

This would be the simplest possible definition, which very few people subscribe. My searches of "God" resulted in far more detailed definitions that align with Christianity. For example, the source you referenced (https://www.merriam-webster.com/dictionary/god) has a primary definition of the Being perfect in power, wisdom, and goodness who is worshipped (as in Judaism, Christianity, Islam, and Hinduism) as creator and ruler of the universe.

On the other hand, few if any people living on earth likely believe that the spaghetti monster has created much of anything.

This is an analogy. What you are doing are choosing ways in which God and the Flying Spaghetti Monster (FSM) are different; analogies look at the similarities between two things. One can compare apples and oranges, in that they alike because they are both fruits. Dawkins, and many other philosophers, use the FMS to compare to the Christian God that comes with a laundry list of supposed properties that are unfalsifiable and have no evidence... just like the noodly appendages of the FSM.

In short, if you Strawman Fallacy one's analogy, you are the one committing a fallacy. If you have any evidence of Dawkins or anyone saying something similar to "a creator of the universe is just as outrageous to believe in as the FSM" then they would be guilty of a Weak Analogy .

There can be no illogicality here because The Flying Spaghetti Monster is itself a god: the god of  the Pastafarians, whose deeply held beliefs must be treated with respect and sensitivity.
See https://www.spaghettimonster.org and https://en.wikipedia.org/wiki/Flying_Spaghetti_Monster

The FSM is not to be compared to the Invisible Pink Unicorn, which atheists claim can substitute the word "God" in any context and make as much sense, eg  "I swear by almighty Invisible Pink Unicorn" or "So help me Invisible Pink Unicorn."
See https://en.wikipedia.org/wiki/Invisible_Pink_Unicorn

Not a False Equivalence nor an Appeal to Extremes on Dawkins' part since "God" and the FSM both exist as non falsifiable premises, therefore both have equal weight to either exist or not.

An Appeal to Popularity and Appeal to Anger fallacies are used by the questioner for Dawkins comparing the Western God to the faux religion of the FSM. I'd toss in an ad hominem attack as well.

The only violation of logic that I see here is in your reasoning. Your conclusion doesn't follow the premises, which is a Non Sequitur Fallacy. 

"On the other hand, few if any people living on earth likely believe that the spaghetti monster has created much of anything."

This is a Appeal To Common Belief Fallacy. Whether many people believe in the Spaghetti Monster, God or not has no barring on its logical accuracy.

The "Flying Spaghetti Monster" is an analogy used to show certain logical fallacies committed by some religious groups.  

#3 is Poisoning The Well Fallacy, and Ad Hominem Abusive fallacies.