So I've been searching for information about the console and just find out that it comes with a code that gives you coins... that you can change for games. 

My question is, if I buy a 3DS and I register the code, It will give me 160 coins right? ok, so after I do that I can go and select a game let's say... Fire Emblem awakening, buy it with my coins and download it? Is that how it works? I was searching at Nintendo's homepage for this and found the list of  Nintendo games that are working with this system and It says that Fire Emblem is worth 60 coins. 

Explain me this a little bit because I just can't believe it... and I'm sure something must be wrong with what I understand.

You have about half of it right. Club Nintendo rewards your new Nintendo-related purchases -- most hardware, most games published by Nintendo, usually not games on Nintendo consoles published by third parties, with some exceptions -- with virtual "coins." You can exchange those coins for certain rewards.

If Fire Emblem: Awakening is "worth" 60 coins, that means you will receive 60 coins if you buy it and register it on Club Nintendo. It doesn't mean you can buy the game for 60 coins! Usually the games on offer are Virtual Console or WiiWare/eShop titles, and they usually cost 150 or more coins apiece. For instance, right now you can "buy" Super Metroid on the Wii Virtual Console for 150 coins, or Super Mario 64 on the Wii Virtual Console for 200 coins.

Also, if you register 300 or more coins (doesn't matter whether you spend them) throughout the year, you'll qualify for a gift from Nintendo at the end of June. Last year, I got Metroid II on 3DS Virtual Console through that program, but there were other options as well.

Hi, no you can't... Fire Emblem gives you 60 points when you buy the game and those points your are going to use them to download few games on the eShop, none of these games are big titles tho...