Hello this is my first post here. I am Japanese and was used to play mah-jong so hope my answer might help
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1) 第一打 is the first round?
No. the first throw of the game to be exact.
2) リーチ一発 meaning "first turn win?" Can be translated as "declare Riichi on the very first turn"?
No. リーチ一発 means a hand that the player wins the game within the first round after declaring Riichi that gives extra points to the player. "Declare Riichi on the very first turn"? is another hand called "double Riichi".
3) 待ち means wait, so it's the game stage when a player have already chosen the tiles and he's waiting for the opponent's move?
Yes.
4) マンズ待ち is something like "waiting for the Manzu". Does it make any sense? What is マンズ, a type of tile?
Yes. Manzu is a kind of suits with Chinese characters on it.
5) 安牌 is used to indicate the safe tiles, right?
Yes.
6) 打つ and 打ち込む means "to throw", so are synonyms of 捨てる and 切る, "discard"?
打つ, 捨てる and 切る all means simply "to throw" or "discard", but 打ち込む means to discard a tile that is picked up by other player to win the game so the word implies "to throw a wrong tile".
7) 河底ロン is read "Houtei Ron", but there is any translation for it?
河底ロン means wining by picking up the very last tile of the game (either from the pile or the discarded tiles).
8) 持つ is the term for "to draw"?
No. 持つ means to keep it within your hand (tiles) in this context.
9) 通し合う has a related Mahjong meaning?
Yes. 通し合う means multiple players to cheat others by exchanging signals.
10) 牌 is "tile"? So, 手牌 could be generically translated as "hand", referred to the player's hand (the tiles he have at the moment)? Or maybe 手牌 is a tile piece? I thought 手牌 was a synonym of 手, but maybe it means "Tenpai tiles", the tiles you have in Tenpai? Or is that only tile you need to
reach Tenpai?
手牌 simply means the tiles he has at the moment.