For example, in the case of the following sentence, do you capitalize the "Good Morning!" and the next part "It..."?
Mr. Smith, Good Morning! It was a pleasure meeting you the other day.
Please let me know. Best, James
The simple answer is no; you do not capitalize "good morning." The correct capitalization is this: Mr. Smith, good morning! It was a pleasure meeting you the other day.
And indeed, although "good morning" doesn't have a subject and verb, it is still considered a complete sentence -- and so, "Mr. Smith, good morning!" would also be a sentence. That is, some groups of words, such as "Good morning" are considered sentences even though they don't follow the rules.
"Mr. Smith, good morning!" is not a complete sentence.
It does look like something I'd expect to see at the beginning of a letter. In a letter, the salutation ("Mr. Smith") and the beginning of the first line ("Good morning") would both be capitalized. Note that the salutation is written on a different line:
Dear Mr. Smith,
Good morning! It was a pleasure meeting you the other day.
To answer the question in your title, we only capitalize the first word of a sentence any subsequent proper nouns.
John Smith Wished me a good morning -> John Smith wished me a good morning
President Lincoln Delivered the Gettysburg Address -> President Lincoln delivered the Gettysburg Address
McDonald's Is a popular restaurant chain -> McDonald's is a popular restaurant chain.
Also note, there is no reason to capitalize "Morning" in "good morning" (unless you're using "Good Morning" as a title).