you can use it,they are same

Yes of course. "Gene" is more colloquial but you can see it in the text as well. However it is not that suitable for formal I think

They are more or less exchangeable, however to a native ear, they are not the same. Yine - and the colloquial form of it "gene" - are original Turkish forms which have a common stem with the word "yeni"(new). In that sense, you can take "yine" to be equal to the word "anew" in English. It suggests a repetitive action but in a new form. On the other side "tekrar" (stemmed from "kere") is an Arabic loan, and it actually means a "repetition" and suggests a routine event take place. Summing up; for example, if you are inviting someone to a party who had already been to your lodge before, it is better to say "yine gel!" since it implicitly means that you really want him/her to come again.

same

I don't think so. "Burası" is more like "this place." So to say the English sentence with "this place" you have to say "Come to this place again." But "to this place" requires a dative ending. That means you have to use "buraya."

Not correct. It should be: "Yine buraya gel" or "Buraya yine gel"