Learn English – Why do common swear words have four letters

is there any accepted/standard list of the 100 most common English words?

I suppose it all depends on your definition of authoritative, but I think a good start is The Oxford English Corpus, a collection containing over 2 billion words of 21st century English from around the world. Here's a list of facts about the corpus, including the 100 commonest words in the English language.

Neat facts about distribution: 10 lemmas (word forms, is and are are lemmas of to be) make up 25% of the corpus, 100 make up 50%, 1000 make up 75%, 7000 make up 90%, 50,000 comprise 95% and you need over a million to get 99% coverage.

So, one quarter of all words used are the, be, to, of, and, a, in, that, have, and I.

Is it a myth that they're all Germanic in origin (as I now doubt)?

Yeah, most of them are germanic in origin, but not all.

As you noted:

use is of Latin origin (by way of French) and replaced the O.E. verb brucan (which survives as the verb brook "to tolerate, put up with something unpleasant")

because is of direct Latin origin from the phrase bi cause "with cause."

and

people also Latin by way of French.

Those are the only words that jumped out at me. Of course, most of the common words have Indo-European origin, so they'll ultimately share a common root anyway. See two and duo.

Let's not forget that Latin was the language of science and philosophy and nearly all higher learning for hundreds of years. For example, Copernicus wrote his seminal treatise about the heliocentric theory, De revolutionibus orbium coelestium, in Latin. It's not surprising that many of the technical names for natural phenomena and educated words in general would be derived from Latin.

We see remnants of this today in medicine: the parts of the body are named from the Latin. Your orthopedist will tell you you've torn your medial collateral ligament in your knee.