One of the key rules for understanding subjects is that a subject will come before a phrase beginning with of. The word of is the culprit in many, perhaps most, subject-verb mistakes.

Incorrect: A crate of sardines are* more expensive than I thought.
Correct: A crate of sardines is more expensive than I thought.

In addition, the words each, each one, either, neither, everyone, everybody, anybody, anyone, nobody, somebody, someone, and no one are singular and require a singular verb.

Each of these hot dogs is juicy.

Everybody knows Mr. Jones.

As for "We each have a laptop", in my opinion, it is a sloppy sentence compared with "Each of us has a laptop" or "We all have laptops".

If the intention were to use "each" as an adverb, it would and should be placed at the end of the sentence: We have laptops (,) each (of us).

And here's the final point:

With words that indicate portions—e.g., a lot, a number, a majority, some, all. etc.— the previous rule is reversed—and we are guided by the noun after of:

If the noun after of is singular, a singular verb is used. If it is plural—a plural verb.

A lot of the pie has disappeared.

A lot of the pies have disappeared.

A third of the city is unemployed.

A third of the people are unemployed.

All of the pie is gone.

All of the pies are gone.

Some of the pie is missing.

Some of the pies are missing.

Here's one of the numerous sources.