[Yağız´ın [Ömer´in oğlu] olduğuna] inanmıyorum!

The genitive case has many functions. The basic function  -  possession  -  can be seen in Ömer’in oğlu ‘Ömer’s son’.

Another function is grammatical. When you make a sentence

Yağız Ömer’in oğlu

a constituent of another sentence like

İnanmıyorum: Yağız Ömer’in oğlu.

(LIKE Sana/Allah’a/Aşka inanmıyorum.)

you must mark somehow that this sentence has been subordinated/embedded and it is not an independent sentence any more. This is where Turkish language uses genitive marking for the subject (and participle marking for the predicate). So,

>>>> Yağız’ın (GEN) Ömer’in oğlu olduğuna (PART) inanmıyorum. ‘I do not believe Yağız is Ömer’s son.’

Please send me the request.

Shall I send more?

It seems that I rewrote the lines. :O

Without the full stop:

No matter how much my nerves test the limits of betrayal

there is still a smile in my secret cache.

If you said laugh I would laugh.

This is interesting. I see some ambiguity, maybe even a word game. This is sort of a try though:

I have such (strong) nerves.

Even if your betrayal tested their limits

I still have some powder of smile in my secret storage.

But if you said laugh I would laugh.

On the other hand, if GÜL was used here as a term of endearment it would make more sense...meaning if you spoke to me sweet words after what you have done I would not take them seriously. But this is just guessing from my side.

The overstroken lines are not grammatical.

The AN participle is used when the constituent governing it is the grammatical subject of the main clause:

kadın geliyor/geldi > gelen kadın

kız okuyor/okudu > okuyan kız.

Notice that this holds true for the passive sentence also:

hasta bakılıyor/bakıldı > bakılan hasta.

Roughly, the DIK participle is used when the word governing the participle is the grammatical object or adverbial of the main clause.

Kız okuyor/okudu. > Kızın (GEN) okuduğunu (ACC) gördü.

Kadın geliyor/geldi. > O kadının kim olduğunu biliyor musun?

Adam şehirde yaşıyor/yaşadı. > Adamın (GEN) yaşadığı şehirde deprem oldu.

diye marks a straight quotation. That is why we don´t need to play with infinitives/participles at all when using it. 

Neredesin diye merak ediyorum.

Nerede olduğunu merak ediyorum.

Or in a manipulative sentence with infinitive:

Burada kalmasın diye istiyor

Burada kalmamasını istiyor.

Actually there are many questions in your question, raydin. Feel free to ask again if this answer was too condensed.

You are not finished when you lose. You are finished when you have nothing left to lose. We are not finished yet.

My Try:

I want to face a person who is more mature than me,

more honest

even in the smallest matter,

so mature that they won´t turn their back

and if I turn mine

trustworthy enough not to stab in my back.

This person should be brave enough

not to fear of getting wet in the rain and not to escape from the storm,

not to stand in fear for a rock after stumbling on a stone,

someone who loves his work

and who doesn´t give up when things get difficult.

Because time expressions are often simplified. Not only in Turkish. Just like in English we say this summer instead of in this summer. One of the reasons is their frequent use.

Turkish grammar is so regular that we learners tend to take every exception in the rules as a personal insult.

Besides, native speakers are often unable to answer our why questions because they are not relevant to them. They just let it go and speak.

I only see one actual mistake here  -  but you repeat it many times. The other remarks are about idioms  -  English or Turkish. Neither one is my native language so they are more commentaries than corrections.

P.S. You have been very careful about translating every IYOR into BE -ING. There is no one-to-one correspondence between these verb forms. But if it sounds natural it is ok of course.

A very sexist joke, tunci.