the split of a sentence

- I cannot find any difference between your first and second sentences.
- The third sentence is definitely wrong. You cannot break down the sentence with a comma between "about" and "asking".
- The first sentence is the best. But "asking for mergin" is too clumsy.
- Also in the first sentence, please change "about" to "to". If you want to use "about". You have to change to "I sent an email about my request on asking....."
Therefore the change I suggest is :
I sent an email asking for merging my 3 accounts to one to [email protected] and [email protected].

I would put it as:
I have sent an email to [email protected] and [email protected]. respectively for requesting a combination of my 3 accounts.

respectively - 分別地
merging - 一般指 (公司等)合併, 收購
combination - 指結合(體);聯合(體)
(所以 merging 個字唔係太合適)
(I sent an email about asking for merging 3 accounts to [email protected] and [email protected].)
係錯 grammar, 首先唔理用詞對或錯, 正常寫法次序都會係 I sent an email to [email protected] and [email protected] , asking for merging 3 accounts.
如果唔係咁寫, 句頭 I sent an email (OK), 但句尾會係 about asking for merging 3 accounts to [email protected] and [email protected] 點解得通喎??

I sent an email about asking for merging 3 accounts .... 錯. 字眼方面須改進, 加上文法亦有些少錯. 每逢這類official notice:
1) 你須用 request i/o asking 記住, ask 是命令式. 2) 須有準確的日子可供查閱.
In grammar: I sent an email, I eat dinner 等都不對, 缺乏時式.
asking for merging 3 accounts... 應改用 requesting to merge the following 3 accounts.... 較為達意.
I suggest you better write the sentence like this:
I've sent an email (today or yesterday or a date) requesting to merge the following three (3) accounts into one...................(details)