So for each due date, the payments are for the previous quarter. Here is a link on Michigan estimated taxes: https://www.michigan.gov/taxes/0,4676,7-238-75545_43715-182761--,00.html
Yes, you will still need to make the federal estimated payments.
If they are willing to withhold, you can have them withhold the taxes for you so you don't need to worry about the estimated payments. To figure out how much to have withheld, you can complete a new W-4 with your company. You can complete a W-4 in TurboTax by going to Other Tax Situations in your return and then going to the W-4 section.
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