Before delving into the meat of the question, I would like to argue that, semantically, "humanity" is probably a misnomer for the condition you are trying to understand. It implies the importance of being a member of the human race, which is limiting to the larger question. Instead, "personhood" is the term I would suggest.
The core of this questions is the "Mind-Body Problem". There are two general sides to this discussion; Dualism, and Materialism.
Often attributed to Rene Descartes, mind-body dualism is the concept that there are two separate but interacting portions of our existence; the physical brain and the ethereal mind. To the dualist the mind is a quality in and of itself, and it interacts with the brain to manifest intellectual understandings. Opposing this view is materialism, which argues consciousness is completely dependent on the physical body, i.e. the brain and the mind is a manifestation of the brain.
I would refer you to Rene Descartes, Daniel Dennett, John Searle, and Thomas Nagel as some good sources to begin framing the two perspectives on the question. Here is a discussion with John Searle that discusses many of the foundational points and counterpoints to this debate.
In response to your first set of questions...
Is a human brain in a vat human?
What about a human brain in a sheep?
Or a sheep brain in a human?
While the dualist and materialist do not agree on the origins of the mind, I believe that both hold agree that it is the existence of a mind that establishes personhood. Descarte described an animal brain as a clockwork mechanism devoid of a mind. Therefore, the sheep's brain in a human's body is an animal.
In the reversal, where a human brain in transplanted into a sheep, personhood could perhaps be transferred too. For a Dualists it would depend on if the mind is also transferred. For the Materialist, the mind necessarily transfers where the brain goes (if the brain maintains its functionality). This is also true for a literal "brain in the vat" scenario.
However, to ask if the mind could be "downloaded" into a computer, rather than transferring the brain to a vat frames the question slightly differently. It poses little issue to the dualist, since the mind is separate from the brain, and the computer would simply replace the mechanical portion of the current role of the brain. However, to a materialist, each mind is the result of a specific brain, therefore begs the question of how the same mind could exist outside the specific brain of its origins.
In response to your last question...
What do the answers to these questions say about what has the potential or is dictinctly human?
John Searle has an interesting thought experiment called the "Chinese Room", which he briefly discusses in the link provided above. Despite being a materialist, he makes the argument that simply making a machine that takes input and creates outputs that would mimic our thought process doesn't reach the level of understanding that is required to describe it as a mind. So, to your question the potential for person in any form is an ability to think and also understand.
Others, like Harry Frankfurt and Bennett Helm, focus on emotionality and choice as the key aspect in the attainment of personhood. Returning to the question of the sheep brain compared to the human brain, Helm argues that the ability have second-order desires and second-order volitions are the threshold that separates an animal mind from personhood.
First-Order Desire: a desire to do something; i.e. a desire to smoke.
Second-Order Desire: a desire to have a first order desire; i.e. a desire to quit smoking.
Second-Order Volition: the desire that one's first-order to be one's will, and to move one to act.
(Emotional Reason: Deliberation, Motivation, and the Nature of Value, Bennett Helm, 2001)
For an animal, there is only first-order desire. Given two desire, the one with the most pull will always win, despite any higher goals or considerations. Or in the case of equal desires, an animal such as "Buridan's ass" (LINK) would be forever stuck. However, Helm argues that a person capable of second-order volition would not be stuck. Instead they could call upon their ability for second-order volition to overcome the deadlock. Thus it is that criteria that must be met to establish personhood, as opposed to an animal mind.
Similar to your questions, another interesting permutation to evaluate is "if I take my brain and put it in your body... am I me, am I you, or are we a new person?"
More question arises from conditions like Dissociative Identity Disorder (previously known as Multiple Personality Disorder). I would think that for a materialist, no matter the number of personalities, a person with D.I.D. is one person. However to a dualist, I believe the question of whether each distinct personality is a distinct "mind" would be complicate the issue, and lea them to say that each personality is a different person.