It's a good question, and unfortunately, the answer is that zipcode does not map well to other kinds of geographical regions, or to day-to-day-reality for that matter.

To directly answer your question, it is not uncommon for zipcodes to span county boundaries. And there are even a handful that span more than one state: https://gis.stackexchange.com/questions/53918/determining-wh...

To the broader question of why zip codes can be lacking for geospatial analysis, here's a real-world situation:

https://theconversation.com/how-zip-codes-nearly-masked-the-...

Here's a more broader essay from CartoDB: https://carto.com/blog/zip-codes-spatial-analysis/

I don't think it's generally possible. The FIPS code just identifies the county (the first two digits are the state, e.g. 06 for California) and the final three identify a county in that state. So technically that column is redundant with the "county" and "state" columns; I suspect they've included it to make joining this dataset to other data that might use different name-formatting of counties/states easier.

ZIP/postal codes are generally smaller (in the county I live in there are almost a hundred ZIP codes). I'm not sure they're even guaranteed to be entirely within one county either. We tend to think of ZIP codes as boundaries but they're actually delivery routes (which if you squint can be converted into boundaries by joining the properties that those routes serve together).

You might be interested in this article: https://carto.com/blog/zip-codes-spatial-analysis/

problem is somewhat ill-posed, but see

https://raw.githubusercontent.com/bgruber/zip2fips/master/zi...

FIPS codes are only specific to the county. It'd be a one-to-many mapping (or many-to-one, depending on if it was a rural area, I guess? I'm not sure if ZIP codes ever cross county lines.)

It's nontrivial.

Check out ZCTA crosswalk in US census.