[yasuto]

Hope I can be of your help.

11) 振り込む is another synonym of 打ち込む? Means to discard/to throw with the nuance "to throw a wrong tile?"
Yes.
12) 振る　is a synonim of 打つ、捨てる and 切る、 right?
Yes.
13) 先打 is the first player to discard? Is different from 先攻?
I am not sure. I have never seen or heard of the word before. It could simply mean a player whose turn to play is earlier than another (後打) player among four. t could also mean throwing a specific tile in an early stage of the game. I googled and found a word 好牌先打, which means throwing a valuable tile in an early stage of the game.
14) 後打 is the second player to discard?　Is different from 後攻?
I am not sure. I have never seen or heard of the word before. It could simply mean a player whose turn to play is later than another (先打) player among four.
15) アガる means "to win" in this context, right? So, for example, "アガれる方のロン牌 is "the tile you could use for Ron"?
Yes.