Can psychiatrists do psychological and neuropsychological testing like psychologists? or no?

Yes, they can but: But they typically don't do such testing, since they haven't had the training in giving and interpreting the tests, that the psychologists have had.

Depends: Much of what a practitioner does depends on their past training, personal interests and time constraints. Much of this testing in time intensive & psychiatrists may prefer to have a psychologist do the testing to manage more patients at one time. Others may choose to do some or all of a variety of tests themselves as a way to better understand the patient.