Your first two examples
another two sentences
other two sentences
These are both valid phrases, but they are used in different circumstances. Another is basically a contraction of "an other", used when the indefinite article would otherwise be used. That is, when the "other" items are not a particular set of other items.
The teacher asked us to write 3 sentences, but I wrote another 2 sentences to try to get extra credit.
It doesn't matter what 2 other sentences I wrote, just that I wrote two more than were asked for.
Three of these five sentences are incorrect. The other two are correct.
If I have five sentences to start with, and three are incorrect, then it's two specific other ones that are correct.
Your third example
another of two sentences
There are ways for this to be a valid phrase, but it's meaning is not similar to the other examples. Furthermore, in the examples I can think of, it needs context to provide an antecedent for another to refer to. For example,
In my essay I wrote one paragraph of six sentences and another [paragraph] of two sentences.
Your final two examples
We need another three chairs[.]
This is a valid sentence.
We need other three chairs (*)
This is not a valid sentence. Instead we might say
We've brought 3 chairs from the store room but we still need the other 3 chairs to seat all 6 of us.
This implies there were originally exactly 6 chairs in the store room.