Why do people keep the first day of the week for worship of God and are they really worshiping the Sun god?

Answer: The day was made for man, not the man for the day. We as Christians go to church on Sunday due to the fact that that's when our Lord Jesus Christ resurrected in victory ......Read Matthew 28:1 and Marcus 16:1

It is not possible to keep the 10 commandments which should be the standard by which for every bible believing Christian should live by, without observing the Sabbath day.

It's so common for people to believe that the Sabbath was only made for the Israelites/Jews which is outlined in the 4th commandment. The 10 commandments were established at Creation and most importantly the Sabbath was established at that time. God blessed the seventh day and rested - not just that seventh day, but THE seventh day.

No they are not really worshiping the Sun god. They are worshiping the one true God, God the creator and sustainer of life. The Bible makes it clear that it doesn't really matter where or when you worship God what is important is that you worship Him in spirit and in truth. So if people meet on Saturday or Sunday or Monday it doesn't matter.