The second sentence would be more correct although it would usually be stated as "I live in Russia but am a native of Syria." The first sentence is not grammatically wrong and would certainly be understood. However, the word "place" would more normally be replaced by "country".

I would say, "I live in Russia but I was born in Syria". Here we are assuming you are Arab. For example, the author, Pearl S. Buck was born in China but she was not Chinese; meaning she was not racially Chinese.

Both are ok. Just replace the preposition "to" by "öf" and be happy.

I'm from Syria and I live in Russia. I'm Syrian and I live in Russia. I was born in Syria and I live in Russia. Any of these will do.