僕に話しかけて下さい？？

日本語勉強中　僕に話しかけて下さい ( Correct saying )

"Please talk to me while I'm studying Japanese." --> 日本語の勉強中に / 僕が日本語を勉強している間に話しかけてください。

"I'm studying Japanese / I study Japanese, so please send me message / contact to me." 

--> 僕は日本語を勉強しています。僕にメッセージをください / 僕に連絡ください。

日本語勉強中　僕に話します下さい　( Why is this one wrong? )

*to say "verb + ください", we use the TE-form of the verb, in this case 話しかけてください(talk to me) / 話してください(tell me). 

*I'm not very sure what you wabted to say with 日本語勉強中, we need more detail to make the sentence clear and natural. If I understood wrong, please tell me. 


話しかけて　I dont understand [ kakete ] part. Please help me :)

--> it's a bit difficult to explain, I think basically "to talk to someone in person who is doing his work / who is talking with other person / who don't notice you / etc, with "Hi / excuse me / etc", talk to someone catching one's attention. 

Good job! Please enjoy learning. I hope this helps you. ¡Mucha suerte!