1. I heard Bengu has a new album.

Bengü´nün yeni albümü varmış. 

2. I asked Sila about Bengu´s strange video.

Ben Bengü´nün tuhaf videosu hakkinda Sıla´ya sordum.

3. She told me Bengu is copying her.

Bana Bengü ona kopyalama şöyledi.

4. Bengu wrote me: I know you like Turkish my music.

Bengü bana yazdı: Türk müziğiyi Benim müziğimi sevdiğin biliyorum.

At least is this one gramatically correct at least?

I wanted to use mış, the inferential form. As in, I heard she has a new album but I didn´t know if it was true. I saw varmış here:

http://www.turkishlanguage.co.uk/varyok.htm

The Inferential Form

The Inferential - It seems that there is/was - is used when the subject has no eyewitness knowledge, it is used for reporting and inference. The Inferential is simply formed by adding the Inferential Suffix -miş as the following examples show. The Inferential Suffix -miş is used for both the Present Tense and the Past tense:
varmış - It seems that there is/was, It seems that there are/were.
yokmuş - It seems that there is/was not.. It seems that there are/were not..

Is the possessive, in Bengü´nün, mandatory in 4?

I wanted to use mış, the inferential form. As in, I heard she has a new album but I didn´t know if it was true. I saw varmış here:

Using inferential mood is correct, however, the problem is in choosing the verb “var”, in terms of of  wording “someone has a new album” in Turkish, we generally use “çıkmak”  -à “come out”

I heard that a new album of Bengu came out. 

[Duydum ki]  Bengü´nün yeni bir albümü çıkmış.

or

 I heard that Bengü made a new album.

[Duydum ki] Bengü yeni bir albüm yapmış.

Here the key word is “a new”, means she had other albums previously.

 If  it was her first album, and you were not expecting an album from that person , and you are amazed that she has an album then you may able to use “var ” -à

 Bengü’nün albümü varmış. -à I heard that Bengü has an album.

Is the possessive, in Bengü´nün, mandatory in 4?

No, it is not mandatory. You can form that sentence also this way :

Bengü bana, “Benim şarkılarımı sevdiğini biliyorum”  diye yazdı.

Bengü bana, “Benim şarkılarımı sevdiğini biliyorum” dedi.

Thank you both!