Merhaba!

This time I have a complex question, that may require a longer explanation.. but maybe not Namely, could you explain to me the difference in using the verbal nouns formed from infinitives (by skipping -k at the end), like: okuma, yazma, öğrenme versus the forms with -dik, like: okuduk, yazdık, öğrendik. For example if I want to say to someone on facebook Thank you for liking my picture (Thanks that you liked my picture). And I may say: Beğendiğin fotoğrafımı için teşekkür ederim or Beğenmen fotoğrafımı için teşekkür ederim. Is there any difference in meaning of these two statements?

#edit

Another example came to my mind: I may help you to learn English: İngilizce öğrenmene yardım edebiliyorum or İngilizce öğrendiğine yardım edebiliyorum. Which version is better and what´s the difference?

(Thanks that you liked my picture). And I may say: Beğendiğin fotoğrafımı için teşekkür ederim or Beğenmen fotoğrafımı için teşekkür ederim.

There is no difference between them. Just need correction. If i were you, i would say "Fotoğrafımı beğendiğin için teşekkür ederim." 

I may help you to learn English

This is an offer sentence, thats why you should translate it like a " İngilizce öğrenmene yardımcı olabilirim."

Ok, thank you But basing on these examples I would like to learn some general rules from you. For example, as you said above "offer sentence" = "only -me form"

It is a big bite, sufler, but I like the way you are going for it.

harp00n cleared the examples for you.

As another learner I can tell you a couple of guidelines which I know  -  but you will have to wait until one of the teachers will give you a proper grammatical answer.

1. Infinitives are often used with verbs which denote manipulation. This means, you will probably find infinitives in sentences with orders, instructions, commands, permissions. Personal participles are used in declarative sentences, i.e. in sentences where things are simply reported and the speaker doesn’t intend to change the listener’s action.

A classical example of this difference is in Lewis’s Turkish Grammar:

Çocuklara aşağıya inip kendisini sokakta bekle|me|lerini söyledi. ‘She told the children to go downstairs and wait for her in the street.’

Çocuklara aşağıya inip kendisini sokakta bekle|dik|lerini söyledi. ‘She told the children that they went down and waited for her in the street.’

2. Infinitives (verbal nouns) stand for action, the fact of action or the result of action. When you talk about the action itself you use infinitives:

Bunu bilmek yeter. ’It’s enough to know this’

As you know English well, I can tell you what I have noticed. Personal participles are used in equivalents of certain types of English relative clauses (…which…), declarative content clauses (…that… ) and indirect questions plus in sentences which you can easily return into any of these three.

Çocuklara aşağıya inip kendisini sokakta bekleme|diğ|ini söyledi. ‘She told the children that they went down and waited for her in the street.’

" Çocuklara , [onların] aşağıya inip kendisini sokakta beklediklerini söyledi."

Ok, thank you guys Maybe this will be enough for me for now, but I have one more question about the example above, but not related to the participles... Çocuklara aşağıya inip kendisini sokakta beklemelerini söyledi. Would it still make sense if I replace the word kendisini with onu = "for her"? Or kendisini is necessary here?

 Yes , you can replace "kendisini" with "onu" in this sentence.

 I agree.

First of all, interesting but not true that me is made by skipping k at the end of mek..

Second, me doesn´t make infinitives.. but maybe gerunds (as in yüzme delisi - fond of swimming) similar to (but not always the same) the -ing form of an English verb when used as a noun.. and sometimes as an adjective (present participle- verbal adjective) (as in bekleme odası - waiting room) I try to explain it by using the English grammar which is not something that I like but I hope it helps..

Third, not all the nouns made from verbs are participles.. 

And I wonder..why -me vs -dik ?

If it were -en vs -dik I would understand it:

(as in relative clauses) 

-en/-an subject pronoun 

Ali ile konuşan kızı tanıyor musun? (do you know the girl who is talking to Ali )

-dik object pronoun

Ali´nin konuştuğu kızı tanıyor musun? ( do you know the girl who Ali is talking to)

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Turks never say "fotoğrafımı beğenmen için teşekkür ederim".. it should always be beğendiğin .. why? because it should indicate a past action ..

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"İngilizce öğrendiğine yardım edebilirim" is incorrect... why? because öğrendiğine makes it look like s/he can already speak English.. it should be öğrenmene.. why? because it should indicate future..