It seems as though the essential question is whether or not the OP is a "Nebraska Resident".
Here is how the administrative code defines the term Resident:
shall mean a person who has been a resident of this state for at least one hundred eighty days. This does not include an applicant who maintains a residence in another state and claims that residence for voting or tax purposes.
So, bigmountain, it sounds as if you are not a resident. Your wife, on the other hand, sounds as if she is.
This is relevant to all three of your questions.
1. Federal law requires an individual to take possession of a purchased handgun in his or her state of residence. To purchase your friend's handgun, you would need to obtain a Nebraska purchase permit, conduct the transaction and then have your friend ship the handgun to a FFL in Utah where you could take possession of the handgun. Person-to-person (non-FFL) transfers must be between two residents of the state.
2. Since you are not a resident of Omaha, you get a pass on the registration on that count (Residents of Omaha must register). Since you hold a Utah concealed carry permit, with reciprocity in Nebraska, it would seem that you also get a pass on that count (Those with recognized concealed carry permits and in compliance with the Nebraska Concealed Handgun Permit Act are not required to register, but my do so voluntarily).
3. Your wife, now as a Nebraska resident (assumed), will need to acquire a Nebraska concealed carry permit. Depending on when it expires, she may want to hold on to her Utah resident permit.
ETA: If you and your wife maintain a (jointly titled, leased, etc.) residence in Utah and she maintains her voter registration in Utah, you might be able to skate on her being a resident of Nebraska. The Nebraska driver's license will muddy the waters, though. Expect considerable legal wrangling if she is found to be carrying.
