The "short" form seems to be developed from the "long" form as a result of phonetic changes.

-e/-i cases and possessive suffixes
gelmeğ-im > gelme-m
gelmeğ-e > gelme-y-e
gelmeğ-i > gelme-y-i

Today "long" form seems to be not used anymore so TDK decided that "short" form should be used.

For genitive, I think it was also similar development for Istanbul Turkish:

gelmeğ-in > gelme-n-in

But take this example:

Kumuk Turkish - İstanbul Turkish
ALLAHU TAALAĞA İNANMAQNI BAYANI - ALLAHU TEALÂYA İNANMANIN BEYANI

As for your examples, I think it's about getting right which case should be used when.

- Bunu Türkçe yazmayı deniyorum.
And NOT (as I would have expected): Bunu Türkçe yazmak deniyorum.
-i case is required here. Nominative case is not used

- Sık sık olması gerekmiyor.
And NOT (as I would have expected): Sık sık olmak gerekmiyor.
Both form can be OK with different meanings:
(Bir şeyin) sık sık olması gerekmiyor.
Sık sık (orada) olmak gerekmiyor.

And there is a difference in the first and second example:

First one is the object of the verb (hence accusative). And the second one is the subject of the verb.

(Ben) bunu Türkçe yazmayı deniyorum. - (S)OV
Sık sık olması gerekmiyor. - SV

Maybe we can get on the same page about terminology and notice that nowhere did I refer to a "LONG infinitive" but rather the "FULL infinitive" and the short infinitive. I would be happy to use the Turkish terms for these two forms, but I don't know them.

Discussing this usage and reading up on it has been a big help to me, clearing up most of my confusion. Before learning this, I thought the /ı/,/a/, etc in dictionary entries just referred to noun objects of the verb.

As for Kumyk dialect, maybe I missed the point, but I will leave Daghestani dialects aside for the time being and just try to get my head around standard Turkish.

Thanks for pointing out the difference in the two examples I was asking about:
[(Ben) bunu Türkçe yazmayı deniyorum. - (S)OV
Sık sık olması gerekmiyor. - SV]

Now let me ask ANOTHER QUESTION about use of the full infinitive and the short infinitive. Look at these sentences:

(1) Yüzmeğe gittim. (full infinitive)
(2) Yüzmeye gittim. (short infinitive)

(3) Konuşmağa başladım. (full infinitive)
(4) Konuşmaya başladım. (short infinitive)

Is there a (yes, subtle) difference in meaning between (1)&(2) and between (3)&(4)?

Ronnie