Lunar New Year

Today is the beginning of Chinese New Year. Gong Xi fat chai!
Almost everywhere you read: Chinese Rooster Year, yes almost. According to one website that I found, it is NOT Rooster, but Hen because it is a Green Wooden Chicken of which has a female character. But we are living in a patriarchal world... So Rooster it is.

Now, there are two things I didn't understand.

First, In Indonesian papers I saw they put 'Chinese year 2556' starts today, while in other websites they put year 4702 or 4703. The explanation of year 4702 and 4703 is the difference of the starting point. The year 4702 came from the old legend about the Yellow King who became king in 2697BC. According to the legend the inauguration of Yellow King was held in the spring of 2697BC. However he used other starting point in his calendar, namely the winter solstice day which was around December 23rd in 2698BC. Therefore some people believe we are entering the year 4703.

So far I couldn't find any explanation on the year 2556. Can anyone help?

Second, They said because on the 9th is a new moon day, therefore it is the first day of the first Chinese month. However, Islamic/Javanese/Balinese calendars which are also lunar calendars marked February 10th to be their first day of the year.

The last time I checked, we only have one moon. So can anyone please explain to me how is it possible?