gönderecekleri temsilci:"The representative who they will send" - "gönder-ecek-leri"  this is an adjective verb. An adjective verb can have possesive suffix "ex: -leri" in Turkish because it can function as a noun like all adjectives. "-ecek" here is not only a tense it is also adjective verb suffix.

konuşabileceği birisi : "Someone who he can talk to" "konuş-abil-ecek-i" this is an adjective verb too. -ecek has the same function here too. The adjective verb has a possesive suffix -i here .

If you want detailed information, you must study "fiilimsiler".

http://www.turkdilbilgisi.com/fiilimsiler/fiilimsiler.html

So in the sentences above there are possesive suffixes not the definitive (?) suffixes.

(?) I dont know how to call them in English

Object in the first sentence is grammatically definite because "bir çocuğu" has the -u suffix but I think it is a mistake to make it definite because "bir" already makes it indefinite.  It must be "Sokakta bir çocuk öptüm." If we want to make it definite we must remove "bir" and add -u suffix after "çocuk":" Sokakta çocuğu öptüm." Sometimes we do this kind of mistakes in Turkish.

"kaptan olacak birini" is an adjective phrase. So it is correct. It means "someone who is going to be a captain"

Here is a start.

"Sokakta bir çocuğu öptüm" here object(bir çocuğu) is definite not indefinite. When you use accusative case (-ı,-i,-u,-ü ) you make that object definite. So function of accusative case in Turkish is to make objects definite. I guess we have a different understanding of accusative case in Turkish.I guess in English grammar if a word is a direct object of a verb it is accusative no matter it is definite or indefinite but in Turkish translations I see that the term "accusative case" is only used for definite direct object. So this creates confusion for me. 

So I will try to explain things without using "accusative" term. Some verbs want its objects to be definite(ends with -ı,-i,-u,-ü ), some don´t. Maybe there is such a list. I am sure some Turkish dictionaries can be useful too.

Kaptan olması için birini seçin. 

Ol-ma-sı. You need possesive sufix -sı.

I really appreciate your enthuaism about trying to understand every episode of this tv series bit by bit with the translations. I guess no force can break a woman´s will if she is really determined to watch her favourite tv show As a man I would already give up if I was in the same situation.

As translated above : Be jealous of me, dude. Ulan is a slang way of adressing someone and it is not kind. I prefered to translate it as dude but dude doesnt seem offensive but ulan can be very offensive. I use it sometimes when I address a very close friend if we are making jokes to annoy each other or when we talk slang.

My advice to learners is: Don´t focus on grammar, focus on understanding. If you do so you will learn most of the grammar automatically in the process.

I am happy that I was able to help you. As you can see in the book, Turkish grammar is so vast and difficult. I dont think anyone can learn Turkish from a grammar book. It would be very difficult. I recommend reading parallel texts. Instead of trying to learn grammar rules, try to learn the expressions. Most of the Turkish people speak Turkish without knowing anything about grammar. They just know when to use which expression without knowing the structure of them. So I think you should do the same. You will be more familliar with the grammar in time.