I am sole owner of S Corp. Car purchased in MY NAME but used in business, can i still claim it as a business asset of the company and do depreciation?

You can't let the S-Corporation claim the mileage, as the S-Corporation doesn't own the car.

but what about depreciation?

The S-Corp can't depreciate property it does not own. The car will not be listed anywhere on your 1120-S corporate return at all.

At best, the S-Corp can reimburse you for actual expenses that you can provide receipts to the S-Corp for. But if it's not reimbursed through an accountable plan, that reimbursement gets included in box 1 of your W-2 and is taxable income. It's not deductible at all on your personal tax return.

Yes, you can.  You will need to write and sign a corporate resolution stating that the S Corporation is responsible for all payments and operating expenses for the vehicle, and that it is in the owner's name only because the corporation could not get financing.  However, if you drive the vehicle personally, you must add an amount to your W-2  for the taxable fringe benefit.  There are three methods to choose from to calculate the amount, but most use the standard mileage rate multiplied by the number of personal miles driven.