To narrow it down, I live between the 1520s and 1570s.

You picked an unfortunate timeframe - if you were a concubine to the sultan during this time period, then you were probably Hurrem Sultan, because from the early 1520s to the death of Suleiman the Magnificent in 1566 he took no other concubines. After that, you have the reign of Selim II (r. 1566-74), followed by Murad III (1574-1595), both of whom did take a number of concubines. Since you're asking about a generic person to be a character in a story, I'm assuming you're specifically not interested in what it would have been like to be a famous figure such as Hurrem Sultan, so I'll be talking about the reigns of Selim II and Murad III.

Both Selim II and his son Murad III sired children with a single concubine (respectively Nurbanu Sultan and Safiye Sultan) taken before they each took the throne, while they were serving as princely governors. This is what Leslie Peirce calls "temporary reproductive monogamy," because the sultans were monogamous before coming to the throne but took additional concubines after their accessions. Each additional concubine adhered to the traditional Ottoman rule of "one mother-one son" such that reproductive activity with each woman would cease once she bore male offspring. Nevertheless, the sons of Nurbanu Sultan and Safiye Sultan were pre-selected for succession to the throne, meaning that the other concubines did not have any chance to have their sons put in power. Nurbanu seems to have ceased having sexual relations with Selim once he became sultan, and settled into position as the chief figure of the harem - Selim may have even married her, we are unsure because only some European sources mention it. After Murad III came to the throne, he remained monogamous with Safiye Sultan at first, but after some prodding from his mother Nurbanu (still head of the Harem), he eventually took on many other concubines and left a whopping 47 children. Nurbanu died in 1583, outside of your timeframe, but it was only at that point that Safiye became head of the harem.

Is my life effected by where I was born or my ethinicity? Is a european concubine treated different to an arab one for example?

I don't have any source that states this outright, but as far as I'm aware all concubines were European or Caucasian. This is just a matter of how the slave trade functioned - the Ottomans had access to immense numbers of European slaves through both their military campaigns in Hungary and the Mediterranean and the slave markets operating out of the Crimean Khanate. There were also slaves of African origin, but the ones which found their way to the imperial palace tended to be male eunuchs.

Would there be a high chance of me being born into it?

You could not have been born into concubinage in the imperial harem, unless you were born into slavery elsewhere before being imported into the Ottoman Empire. Slavery wasn't an inherited status in Islamic Law. Most slaves would have been captives taken in warfare.

Does my level of comfort and limited freedom depend on how close I am to the Sultan? Finally, how invovled would I be in his affairs? I've heard accounts of some concubines running the empire in all but name.

Yes, your level of comfort would depend on that - kind of. The thing is, the type of character we're working with here would have had a fairly limited range of action. Hurrem Sultan, Nurbanu Sultan, and Safiye Sultan all had immense power and influence. Toward the end of her career, Hurrem had a daily allowance of 2,000 silver coins (akçe), which meant she had access to financial resources equivalent to one of the empire's major provincial governors. Nurbanu Sultan's stipend was 1,000 akçe per day. In contrast, other random concubines who bore sons seem to have been given only 30-40 per day. Certainly enough to live quite comfortably, but absolutely nothing compared to the sultans' favorites. They did not have political power. The harem during this period was dominated by the aforementioned women, and there was no real breathing room for anyone else to exercise any influence.

That being said, there's one form of influence that would be available to you - and that is closeness with Hurrem/Nurbanu/Safiye. Just as the sultan ruled over his male household, these women ruled over the harem, and the women within it were under their command. Your character could gain in status and influence through closeness to them.

All information comes from Leslie Peirce's The Imperial Harem: Women and Sovereignty in the Ottoman Empire, particularly the beginning of chapter 4. I highly recommend this book if you want to understand how the Ottoman Harem functioned.