Need suggestions for Nice activities

I can't help with most of your questions, but:

• The easiest way to get to Monaco from Nice is to take the train. No worries about parking, which I imagine is quite difficult and costly in Monaco. Unfortunately, there's no train service to Eze, so your options are rental car, bus, or a tour.

• Google for Nice's average water temperature in May. I suspect it isn't as warm as you're assuming it will be.

Well you have a party of 8 and it seems only 2 full days so there will be some challenges.
Your list of things to do is short and manageable though.

If all you want to see is Monaco and EZE with the car ; that is not a good idea to rent one.
Plus I think you would need 2 cars for 4 adults and 4 children three of which you will need to rent extra child safety items for. That will be an expensive rental for sure ; but I imagine pales in comparison to the cost of a yachting excursion for 8!

Much better to use public transit for these 2 places and much cheaper. Monaco has parking underground but it will not be free and I imagine you will have 2 cars.
You could after Monaco take a train to Eze sur mer (sea level) and then take a short bus to the top of town, Eze village which is what you want to see. From there a bus back to Nice or the same short bus ride down to the sea and train to Nice.

If on Saturday I think a better plan is probably to bus to both since the before mentioned Bus to Eze from Nice goes right to the top village and you could catch another bus from that same village to Monaco ; but believe I read the bus schedules may not allow that on Sunday where the train and only needing that short bus ride in Eze would not matter Sat or Sun

Lastly, May weather is unpredictable I imagine so if your Saturday not a good beach weather / yachting weather day you would want to switch those.
Therefore I would rather keep which day to do which open and not reserve anything in advance.

Here are a few suggestions:
Day 1: As you are arriving late, if you do need to get lunch, you may need to go along the Promenade, in a hotel or in Old Town to find a restaurant "non-stop". Many of the tastier ones take the traditionally French service break after 2:30. Head up the hill before sunset for some lovely colors. I'd reserve some time one day to the Cours Saleya market in the morning.

Day 2: Antibes or nearby Juan Les Pins have sufficient beaches. It may not be a beach-worthy day in mid-May. You can charter a yacht in either Antibes or Nice. As an alternate, there are some ferries and boat tours that depart from there, and a private charter might drop you back in Nice. Another alternative is Cannes, which also has yacht charters. The train is a convenient method to arrive.

Day 3. Eze and Monaco are reachable and frequently combined. You'll need a van or two vehicles, so reserve in advance. Single day rates are relatively expensive in Nice. There is plenty of pay parking available in Monaco. You will want to see the Old Town/Palace and the Casino/Monte Carlo square (even if you are not going in, an expensive drink in the Cafe de Paris in front is the checklist experience -- and oogle the collection of fine cars parked in front of the Casino). There is also an Oceanographic Museum (for the kids).