"Bindigimiz tren= the train which we got onto"

The tranalation is correct.

Bindiğimiz trenin makinisti : The operator of the train which we got into

Bindiğimiz trende herşey daha güzeldi : Everthing was better in the train we took before

Depends on what you want to say.

There are two possible translations:

Bağlantı kuruyor
1. "(he/she/it) establishes connection" : something tries to get connected. It could be a modem of something like that which actively does the action of establishing connection. It could be also a messanger program.

2. "(he/she/it) finds a connection (between two things)"

Bazı insanlar Marilyn Monroe'un ölümü ile onun Kennedy'nin sevgilisi olması arasında bağlantı kuruyorlar.

"Seninle olmak için bir başkasının kalbini kırdığım halde neden beni reddettin?"

There is nothing wrong with the text. There are a few tiny mistakes but in general it is a good Turkish and the sentences are beautifly and strongly build.

"Althought there are lots of people around me I don't know why I have a feeling like this but for the whole of my life I don't remember feeling so lonely. Maybe this is part of life, maybe it is just temporary but enough is enough. The best time I could ever share with somebody I have lived and will continue living with you."

Both are uncommon. These kind sentences don't have a wide range of use. Rarely this structure is used maybe in such a sentence:
"Sen geldiğinde ben gitmiş olacağım".

Since it is common to see the structure "verb + olacağım" (or any version of it) in translations it doesnt mean it is common in Turkish.

Neşet Ertaş's music is Halk Müziği (Turkish Folk Music). Songs of Halk Müziği are not called 'şarkı' but they are called 'türkü'.

I would never dare to listen recent songs of Sezen. I'm well experienced on meeting ex-lovers again after many years. It is not a good experience. I'm very happy with her image in my mind as what she used to be.

Yes thats right.

I know that you are a fan of Sezen Aksu. The following are her best albums. Very few albums in Turkish music history are comperable with any of these albums.

1978 serçe
1980 sevgilerimle
1981 ağlamak güzeldir
1982 firuze
1984 sen ağlama
1986 git
1988 sezen aksu 88
1989 sezen aksu söylüyor

Yes she is famous. I mean every person in their 30's would tell you her name when seeing her picture or hearing her song. Of course she isnt as popular as she used to be. Her best song I think is "kar taneleri".

Boşvermek: not to care about something
boşver + miş + im : verb stem (boşver) + reported past tense suffix (-miş + first person personl suffix (-im)

Here you can see tenses in Turkish.
http://www.turkishclass.com/forumTitle_6_2565

Using reported past tense for yourself makes it more dramatically talking about the action.

It is a nice song by the way. I like Nilüfer as well. The best songs she song are the old ones mainly when she was working together with the musician and composer Kayahan. Here last albums are just worthless.