Trying to find out if phrases like “went and got” are correct, e.g.:
She went and got the book.
Absolutely. “Went and got” match and can therefore be used together. As a simple test, you can try “she went,” and also “she got,” so putting them together is fine. If you said: “She went to the store and got the book,” you probably wouldn’t even question the phrase. Dropping “to the store” is fine.