Your questions answered:

Why can the seal only live in areas surrounded by the Caspian sea?
It is not known exactly why Caspian seals are confined to the landlocked Caspian Sea. One inference is that the seals came to the Caspian Sea in the Quaternary period* when there were ice sheets were developing. *The Quaternary period is the"most recent geologic period: the current period of geologic time, beginning 1.6 million years ago and characterized by the appearance and dominance of humans" (http://encarta.msn.com/dictionary_1861697935/Quaternary.html)

Are there any laws that protect the seals?
Protective measures for the Caspian seal began in the 1940s with the banning of seal nets. (4)

you said that the seal was an ice breeding animal. What does this mean?
This means that they are born in on the water ice-field in the northern part of the Caspian Sea. ( 3)

So far, has this Caspian Seal Conservation network been successful in everything its done? Has it truly helped the population of Caspian seals flourish more?
The Caspian Seal Conservation network is doing research about the Caspian seal. There has not been an increase in the population, but they are hopeful. They use this information to educate the countries so they understand what needs to be done to save the Caspian seal. (1,2,3)

Does the U.S. have any of these organizations?
No, they United States does not have any of these organizations because the Caspian seal only resides in the Caspian Sea.

Does the Darwin Initiative help any other animals besides the caspian seal?
The Darwin Initiative helps many animals. Since the starting of this organization, they have funded 644 projects including the Caspian Seal Project. (http://darwin.defra.gov.uk/)