You really shouldn't need to fact check everything. Otherwise, you'll never get anything accomplished.
Sure you should.
Fact-checking doesn't necessarily, or even usually, require you to go back to first principles the way you suggested with your baseball example. It requires that you find a credible source (or, preferably, multiple credible sources) verifying the account as true. In the case of your baseball example, if your random friend told you a random stat about a player, you would have reason to doubt him, but if MLB's website says the same thing, you know that it's probably true.
But then you're not fact checking everything, since you're now taking that "credible source" as fact. ---