Please excuse the length of this and you did raise five points, some of which are tricky.
That passage is not a single sentence; it’s three sentences, which read as though they’ve been artificially constructed to illustrate a point…
Whether Stanley has been racing bikes since college is irrelevant both to the meaning and to the grammatical structure.
One. In this case of direct reported speech yes, the full stop (only ever ‘full-stop’ in adjectival hyphenation…) after Stanley’s first phrase is correct. That is because Stanley used not one but two clearly separate sentences.
Different punctuation would be needed if Stanley’s actual words formed a single sentence such as “It’s fun to get out on a Sunday and race and/because/when our team always wins.”
To split a sentence like that would need a form such as “It’s fun to get out on a Sunday and race,” said Stanley, “because our team always wins.”
Two. It is not only OK, it is necessary to capitalize "Our team", precisely because that second quotation is a sentence complete in itself, not a continuation.
Three. In US or any other form of English, quotation marks are clearly the most important difference between reported speech and narrative. The second is the way commas and full stops behave when they appear beside quotation marks.
Note the punctuation when we consider by itself the sentence “It’s fun to get out on a Sunday and race.”
Note the punctuation when we consider the sentence “It’s fun to get out on a Sunday and race” by itself.
Whether or not Stanley said it, the sentence is complete and will in narrative have a full stop of its own, but direct reported speech changes the rules.
My report that Stanley said “It’s fun to get out on a Sunday and race” is a different thing than my statement on my own behalf that it’s fun to get out on a Sunday and race. In both cases, careful wording allows either commas or full stops or neither.
Your comma is appropriate after "and race” because of the quotation marks. Without them there are two separate sentences, even though they have similar subjects. [It’s still a full stop (only ever ‘full-stop’ in adjectival hyphenation…]
Sunday racing fun has no grammatical and little semantic effect on whether “Our team always wins a prize”, which is also complete in and of itself.
Four. That you say ’a new sentence starts after "prize" reveals some cloudy thinking about either the detail or the title of the question, or both. If we were really talking about “a single sentence” then no “new sentence” could start anywhere.
That a new sentence starts after "prize" is relevant only as a corollary of ‘the first sentence ends with “prize”, which matters because which way we think about them influences the way we see any answer. The point is that the old sentence ends with ‘prize’, which is not the same thing. (Please note that strict rules make you and me both wrong to put that as “’prize’,” with the commas outside the quotes and that’s such a different question, anyone interested is welcome to ask about it separately.)
This next is up for debate and my view is that Stanley has been racing bikes since college changes so much, it means that not just a full stop but a new paragraph is needed.
Broadly, in narrative a paragraph is a group of sentences sharing a related theme and in dialogue, a new paragraph is needed each and every time the speaker changes. Also, related sentences should maintain a match of number, tense, voice and most other attributes.
First the example reads as though “…said Stanley” and “Stanley has…” are spoken by different voices, which is equivalent to a change of speaker.
Second “Stanley has…” appears as an original statement by the narrator, while “…said Stanley” is a report. Although the speaker does not change, the use does.
Third both Stanley’s statements express thoughts of or by Stanley; the narrator’s statement expresses a thought about Stanley.
Some combination of that trio seems to mean there should be a new paragraph before the narrator states that Stanley has been racing bikes since college.
Five. That comma used after "and race” at the end of the first reported should not be used after the second and that is not because of consistency, or a lack of it.
The original sentence “It’s fun to get out on a Sunday and race” lost its full stop when it became - if you like, when it was demoted to - a subordinate clause or phrase within the very different sentence ‘ “It’s fun to get out on a Sunday and race,” said Stanley.’
In or out of quotation, “Our team always wins a prize” merited its full stop until I changed it into a subordinate by quoting a quotation… (how many quotation marks belong there is another thing.)
Grammatically, there is nothing like the same relationship between “Stanley has been racing bikes since college” and “Our team always wins a prize” as there is between the quotation about Sunday race fun and the fact that Stanley said it. They merely happen to appear next to each other; there is no question of consistency and from other aspects, they follow different rules for commas or stops.