The most natural would be:

Ça fait deux ans qu’il n’a pas plu dans la région.

or simply (if you are in that region):

Ça fait deux ans qu’il [n’]a pas plu ici.

Il [n’]a pas plu depuis deux ans ici.

If you are not there:

Ça fait deux ans qu’il [n’]y a pas plu.

The most natural sounding depends on the context…

I will distinguish here :

• a written context, where one thinks about the sentence before putting it on paper. All sentences starting with “Il” will fit in this category
• an oral context, where one needs to send out a message rather quickly. Sentences starting with “Cela” or “ça”

Moreover, I would distinguish in parallel a level of language :

• “street language”, or “normal language”.
• formal language expected from higher educated people
  Typically, “ça” is more street language than “Cela”. “Il ne pleut plus…” or “Il n’a pas plu…” remains classic, but flat compared “Il y a deux ans…” which is more poetic in written context. “Il y a deux ans…” would sound ridiculous in oral context. These differences are critical to the french speaker, especially in France (and Europe) and of lesser importance elsewhere.

This being said, present and past do not mean the same thing. From my early education, backed up with (http://bdl.oqlf.gouv.qc.ca/bdl/gabarit_bdl.asp?id=4218) and (http://bdl.oqlf.gouv.qc.ca/bdl/gabarit_bdl.asp?Th=2&t1=&id=4658), past (passé composé) refers to finished actions, whereas present (présent) refers to an action still running. In other terms, your sentences when using present mean that the rain has not fallen for the past two years and it might not change, because the area is a desert for example. Whereas if the sentence uses the past tense, you imply that the era of drought lasted two years and it is expected to end.

I hope this helps.

Here is an analyse of each of your propositions:

“Il ne pleut plus depuis deux ans dans cette région.” is correct. The slight difference between the present and the past is that the present contains the idea that the situation is not supposed to come to an end shortly.

“Il n’a pas plu depuis deux ans dans cette région.” is correct and a rather written style. It has the same meaning as the one above, but does not presume anything for the future.

“Il y a deux ans qu’il ne pleut plus dans cette région.” is correct, but of more spoken style. This variante implies the same idea as the first one, but still stronger, as if it supposed that it would never rain here anymore.

“Cela fait deux ans qu’il ne pleut plus dans cette région.” is correct and written style, because “cela“, is orally used only by high level speakers, the more popular ones using only “ça“. Same idea as the former one, because “cela fait deux ans” and “il y a deux ans” are strictly equivalent.

“Ça fait deux ans qu’il ne pleut plus dans cette région.” is correct. the meaning is equal to the former one, but said in spoken language (“ça” instead of “cela”).

Il y a deux ans qu’il n’a pas plu dans cette région. is correct. The use of “il y a deux ans“, like “ça fait deux ans” is more conversational than “depuis deux ans”

“Cela fait deux ans qu’il n’a pas plu dans cette région.” is correct and belongs to the written style because of “cela”.

“Ça fait deux ans qu’il n’a pas plu dans cette région.” is correct and spoken style.

To finish, we could discuss the use of “dans cette région“. Though it is perfectly correct, on the spoken plan it would be used only by the high-level speaker, the ordinary one would rather say “par ici” if the region about which he is talking is the one where the conversation takes place.

So the perfectly popular spoken sentence would be something like:

“Ca fait deux ans qu’il n’a pas plu par ici“
or even “qu’il a pas plu“, but in that last case, there is really a mistake.