If you stay at Cobalt Coast and you plan to do some off-campus activities you will really need to rent a car.
CC is far from just about everything. It is in West Bay on the west side of the island, north of Seven Mile Beach. It is actually up near Hell (literally, the town/postal code named Hell) but there is no reason to go to Hell unless you want the postmark so that you can "send someone a postcard from Hell!"
Cayman does have a good bus system but it primarily serves the George Town and 7MB areas. It does service more remote areas but less frequently and I believe that the number of buses to outlying regions decreases at night. If you don't want to rent a car the entire week, you can take cabs, but I think the car rental will work out cheaper and more convenient overall.
I believe CC will transfer you to the airport, so you could wait until you are settled a couple of days before renting a car.
Car rental is usually pretty reasonable on GC, but make the reservation in advance since you plan to travel during the high season. Find out what insurance you have on foreign rentals and what your credit card covers - they usually cover collision and theft but not liability, so don't be railroaded into buying more insurance than you need. You will have to show your drivers license and pay $20 for temporary Cayman drivers permit, but they will take care of that at the rental agency.
Traffic on GC can be bad around GT and 7MB mostly during rush hours, but you will mostly be driving in the afternoon and nights so it should be fine. The roads aren't always marked well and the roundabouts can be confusing, and there are narrow-one way streets in George Town. You will get the hang of it pretty quickly, especially if you avoid the heavy traffic periods, but it can be confusing at times.
You should download interactive Google Maps for the island BEFORE you arrive so that you won't need to use data while you are there. We actually prefer to bring a Garmin GPS and car charger with us. Cayman is included in their usual North American maps and we like the way that the GPS talks us through the roundabouts and helps us to select the correct exit.
Rum Point is on the North Side and it is about as far away as you can get from CC without a boat! It will be a good hour to an hour-and a-half drive depending on the time of day. As you head East, after you get passed Bodden Town there is very little traffic. The Czech Inn in Bodden Town is a roadside place with great food. It doesn't look like much but the owner is the former head chef for some of GC's more prestigious restaurants. And their hamburgers are great!
Home - Czech Inn Grill
On your last day before departure, when you won't be doing much diving, you may wish to take a drive to the East End and North Side including Kaibo and Rum Point, stop at Tukka for Happy Hour and watch the frigate bird feeding:
Feed the Frigate Birds! – Tukka
You might also want to stop at Beach Bubbles in Bodden Town and Pure Art east of George Town to buy some souvenirs:
Beach Bubbles Natural Cayman Soaps
Original Paintings, Household Items & Local Art in Cayman Islands
If you prefer to head over to the North Side without driving the entire way, there are options. Red Sail offers a variety of cruise options from 7MB over to Rum Point, including a dinner cruise on Tues and Sat, see:
https://www.redsailcayman.com/evening-sail-west#category/676
If you want to sail over to RP more cheaply ($20 round trip) take the ferry from the Camana Bay Shopping Center not far from 7MB:
Set Sail to the Eastern Side of Grand Cayman | Camana Bay
Cracked Conch (indoor and outdoor patio, more upscale) and Macabuca (outdoor at the edge of the water, more casual) are located on the same piece of property and they are not far from CC and the food is great. It is also a popular shore dive site with a dive op onsite, but you will have shore dive options at CC (weather permitting), see below:
Home
Live web cam at Cracked Conch and Macabuca:
Live Web Cam
If your non-diving spouse wants to do some excursions on her own, I am sure that the folks at CC can help her make arrangements and she could take a cab. There are a number of spas on GC and she might like a spa day while you are out diving. If she would like a beach day on 7MB, she should go to the Royal Palms - but check the number of cruise ships that will be in port that day, because Royal Palms can get packed with cruisers from around 10 am and 3 pm:
Royal Palms | RPBC cruising