Salam, In my opinion, Iraqi dialect is the probably the closest to classical Arabic(MSA). I also noticed that a lot of the grammar rules that exist in MSA are also respected in Iraqi dialect. That's not really a surprise given the fact the 2 of the most influential schools of Arabic grammar are both from Iraq (Al Basra and Al Kufa). As Misslinda mentionned in her comment, Syrian dialect is also relatively close to MSA, but it is much less the case of Yemeni and Egyptian dialects. Anyway, as both Omar and Taleb said, if you try to learn MSA first you should be able to easily learn most Arabic dialects.

Iraqi dialect is a part of Gulf dialect and It's not close to MSA

Kind of but you need to learn at first the MSA so that it can be easy to pick any dialect you want.

Relatively, yes it is close but not alone. Also, in Iraq there are many accents. Which one you want to study. In my opinion, Baghdadi//Iraqi accent is the best among them. Anyway, it is better to study Fusha and then you can choose the accent you want.

relatively yes, but it is complex to start learning a dialect. it will be easier to start with MSA first