You do not require a transit visa as long as you plan to stay airside.
However getting a visa for a quick tour of the city will not be a big hassle and you can even apply for one on Etihad's website. (Passengers flying Emirates can apply on theirs)
TWOV (Transit Without Visa):
Passengers with a confirmed onward ticket for a flight to a third country within 24 hours. They must stay in the international transit area of the airport and have documents required for the next destination.
Visa is required for transit between Dubai International Airport (DXB) and Dubai Al Maktoum International (DWC).
Source: Timatic Via Emirates Visa Check
Do we have to re-check our baggage in Abu Dhabi, as well? If we do, would we require a visa
No, if it is a single booking you do not have to collect and re-check your baggage.
You will go through the US Immigration processes at AUH because Abu Dhabi is a preclearance port, that means that when you reach the US you will not have to go through any more immigration procedures.