Learn English – I received [an A / As] on the exams

Each of my three pens is green is correct.

You're thinking of each of as something that gets tacked onto the beginning of the noun. But when you add each of to my three pens, my three pens becomes the object of the preposition of. Each becomes the head of the subject. So the verb agrees with that, and each is always singular.

Of pretty much always does this:

The rumors were that she would leave him.

The gist of the rumors was that she would leave him.

Some nouns can be plural if they are followed by an of phrase with a plural object:

A number of flaws / A bunch of flaws / A total of ten flaws was found. (iffy)

A number of flaws / A bunch of flaws / A total of ten flaws were found. (ok)

but each isn’t one of them.

The meaning of each X is something like, “This sentence is true for every individual X”. This is clearest when you look at a sentence that is true of each individual, but not necessarily true of the group as a whole:

The roses cost $24. Each rose costs $2.

The phrase in order to connotes enabling

I work in order to have the means of survival.

The phrasing is fairly stiff in most cases, and the same concept can usually be expressed with to

I work to have the means of survival

The phrase for the sake of indicates benefit or support

I made my comments for the sake of harmony.

My comments won't necessarily cause harmony, but should help.

There are cases where either construction might work, but there are at least subtle differences. I don't think that there is a temporal distinction.