In the case of a primary, the date is selected by the states. However, the parties put in place rules that affect the number of delegates a state gets based upon when the primary is. That is how they prevent, say, New Jersey, from moving its primary to the week after New Hampshire.
A state party can hold a caucus whenever it wants. However, again, national party rules restrain states from becoming over eager.
After the 1968 election debacle, the Democrats changed their nominating process for 1972, spreading out the candidate selection process. For a wide variety of reasons (politics, the complexity of the Democrat's process in Iowa), Iowa was made first.
how the order/sequence of states decided?
will Democratic Party follow the same order?
In general they are the same but no always. For caucuses the party is in control of the dates.
does Democratic Party need to start the primary voting on the same day as Republican?
That is entirely up to the states. Do they want to pay for separate elections?