Gersten's conjecture is known for regular local rings containing a field (Panin extended the result of Quillen for smooth schemes). In mixed characteristic, it is known that the statement for a discrete valuation ring implies the statement for smooth local rings over a discrete valuation ring (by work of Gillet-Levine). And the case of a discrete valuation ring is known with finite coefficients (by Gillet away from the residue characteristic, by Geisser-Levine at the characteristic). Unfortunately, rational coefficients are completely unknown as far as I know.
In particular, the answer is "unknown" to both your questions, in mixed characteristic.