1) Re completely white wannabes: I'd say they'd have to be widely or generally accepted as Indians by Indians to qualify. A few Indians saying their white colleague qualifies as an Indian wouldn't be enough.
Take someone like Ward Churchill...please. Since he fails to meet the other criteria, it comes down to whether the majority of Indians accept him as Indian. Ten or twenty years ago, they probably did. Now they probably don't. Hence he's lost the Indian status he once had.
2) Re white men who "married into the tribe": Where's Russell Bates with his tales of the red-haired Kiowa boy when you need him? <g>
I think this case is representative, so I'll post it here:
Frank Baldwin captures mysterious "white Indian" Tehan
So the answer is yes. Membership in a tribe is a political decision, not a racial one. Any government, including that of an Indian tribe, can choose to make whoever it wants a citizen.
I'd strongly advise tribes not to accept anyone, including pure-white wannabes, as members. Because that would dilute their strength as groups with cohesive cultures. But in theory, yes, they can do it.
For more on the subject, see The Facts About Tribal Sovereignty.
Below: Tehan the red-haired Kiowa in Dreamkeeper (aka Archie Morris in ER).