"This letter was written by a man."

Translation: Ten list został napisany przez mężczyznę.

[Webb.Paul]

Wondering why sentence 3. "Ten list była napisany przez mężczyznę." is not also acceptable?

[Jellei]

Because it uses a feminine version of "was". "Ten list był napisany przez mężczyznę." would be okay.

Gender is not visible in present tense verbs, but it is in past and can be in future compound.

[Webb.Paul]

OK thanks.

[hwodzak]

Is it wrong to say "napisany był" rather than "był napisany"?

[0hrstoepsel]

Would "został napisany przez mężczyzn" be correct?

[magpie_gir]

No. It wouldn't.
English article "a" is abbreviated form of "an" (now used only before vowels). It's short form of "one".

"one" can never be equal to "more than one" (plural)

"a man" =/= "men"

[Baerchen_]

So there is no difference between using został or był ? Both translate to the same sentence, if I see this correctly. Which one is more popular and are there any rules when to prefer one version or is it totally interchangeable ?

[Jellei]

I think "został" is a lot better with a participle from a perfective verb (napisać -> napisany), but we somehow can't reject "był". "był" makes more sense with imperfective (był pisany = was being written), where "został" would be wrong.

[MarjanvanK]

what about : ten list był napisany przez człowieka ?

[alik1989]

In this sentence we really need to stress the fact that it was a man, not just a human being who is likely to be a man.

'Ten list był napisany przez człowieka.' sounds like we want to convey that it wasn't written by an alien.

On a side note, even though był is accepted, I recommend został, because it fits the perfective napisany better.