Welcome to the boards Christian2,
You ask some deep questions the early Christians pondered over. Rather than respond to each of them, I'll attempt to provide you with some insights into the theology surrounding Christ's natures which should help you with your questions. It seems you tend to apply an approach to Christ, which eventually became known as the Nestorian heresy
after Nestorius, who is aligned with Antiochene thought.
The Antiochenes saw Christ as having to two essential natures, one divine and one human, viewed as two separate substances being conjoined together. An example would be if you put water and oil in a cup—the oil sits on top of the water and the two never become mixed. You have the two substances together, but they retain their distinction. Replace the oil and water in the one cup with Christ's humanity and divinity and you have the way Nestorians pictured Christ. So when it came to worshipping Christ, the Antiochenes tended to believe we worshipped his divine nature only. To illustrate this unorthodox thought through way of a diagram:
<pre> Human Divine
x ^
\ /
Christ
^
Worship
^
Us</pre>
On the other side you have the Alexandrians. They believe that there was one essential nature (i.e., underlying reality) which had two substances mixed into one. As an illustration, salt water contains two properties (salt and water) and the salt is dissolved into water. In thus way, they would say that Christ's humanity and divinity become mixed together to form one essential nature, and thus they can't be separated (although this is where the salt water analogy tends to fail). It is difficult to get across all the terminology of the original language used in early discussions, but the Alexandrians would essentially say that it is the Son (one underlying reality) who is worshipped, who a mix of two full natures that can't be separated. So when you worshipped the Son, both natures are apart of Him, and so it is not a matter of worshipping one nature or the other. Thus, Alexandrian thought is unlike the Antiochenes' which entails when someone worshipped the Son, they were actually worshipping His divine nature.
Eventually, at the Council of Chalcedon, the
Definition of Chalcedon was formed to resolve this issue. I've attempted to provide you wish some insights, but really it gets deeper and the fact I'm trying to use English to explain the terminology used to refer to Christ's natures, essential nature, etc makes it difficult to get across.
Going onto something slightly new which is relevant to your question, is the word
theotokos—a word Cyril (an Alexandrian) taught in reference to Mary. This word means "bearer of God" and so Cyril was using it to say Mary was literally "Mother of God." Cyril's
theotokos is an outworking of another accepted phrase, the
communicatio idiomatum. It was accepted that 1) Jesus is fully humans, and 2) Jesus is fully divine. Thus, the
communicatio idiomatum was formed to designate that which was true of the humanity of Jesus must be also be true of his divinity, and vice versa.
So some examples would be:
<blockquote>Jesus Christ is God.
Mary gave birth to Jesus.
Therefore Mary is the Mother of God.
Jesus died on the cross.
Jesus is God.
Therefore God died on the cross.</blockquote>
Nestorius obviously didn't like this term as he thought it denied Christ's human nature (as remember he saw Christ as two natures conjoined, which essentially means each nature would retain their separate identity in Christ). Nestorius therefore offers up words of his own
anthropotokos ("bearer of humanity") or even
Christotokos ("bearer of the Christ"), although his denying
theotokos didn't go down too well and so his own phrases weren't accepted over Cyril's
theotokos.
Anyway, I hope what I've written may have helped you regarding how to see Christ's natures and so forth. I'll end here.
Kurieuo.