Is it wrong to put 了 at the end?
Yes, it would translate to something like "has thrown".
丢 is more like "lost" or "dropped", not "throw away"; this sentence almost requires a 把 structure:
我妈妈把我的游戏放在垃圾 or 我妈妈把我的游戏扔掉了
How does 丢 equate to threw away? Shouldnt it be something more like 仍在垃圾桶
Yes, although in English, "throw away" is rather colloquial and metaphorical; 扔 is literally to throw something, like a ball.
我媽媽丟了我的遊戲 are not quite clear what kinda games. In Chinese sense, it has to be describe very clear such as video game 電玩 or table game 桌遊. Game 遊戲 means too general.