Is filing a state W4 form necessary if you already filed a federal W4 form? Do most employers require employees to manually fill out a state W4 form?
There is no guarantee that the numbers you use for the Federal forms will equal the numbers for the state forms. Thus you should fill out the state version of the W-4. It is also a good idea if your state taxes some items differently than the federal government does. If you are getting a big refund in one, or paying a lot of money to one taxing authority each year, it makes sense to adjust one set of numbers.
Every employer I have worked for has required each employee to submit a state W-4 form.
That would depend on the laws of your state. But the states I've worked in have all had their own W4 form, which does not look at all like the federal form, and does not necessarily result in the same numbers.
So yes, if the state says that you are required to file a "state W4" (or whatever they call their form), then you should file it.
As a counterpoint to the other answers, prior to this year my state had never had a state version of the W-4 form, and my employer applied the withholding allowances from the federal form for state withholding also. (That changed this year, because of the significant changes to federal tax law and to the federal W-4, so a state form was finally created.)