Hi anarchy, might be able to answer a few of your questions. We haven't stayed at the Dome but are familiar with the area and are staying nearby at the Macronissos Village later this year. The beach and the area would be fine for children, you would be a 10 minute car or bus journey into Ayia Napa centre which is also fine for anyone most of the day- nice harbour area and Venetian monastery. Please be aware that Cyprus can be exceedingly hot in July or August with temperatures around 40c and Egypt is even hotter if you are thinking of doing the trip. Better to go outside of these times if you can. We did the day trip by air and it was superb, but we did it in spring and from Paphos. I wouldn't have thought it suitable for kids though. Personally I wouldn't do the cruise for the following reasons- I don't like boats, it is a bigger chunk out of your holiday(especially if you have paid AI), and there is a long coach transfer from Alexandria to Cairo and back giving you less time in Cairo itself.
There are other trips available, usually cheaper at the agencies than tour operators. Troodos mountains, Jeep safaris, Famagusta and Kyrenia in N. Cyprus etc. You can book numerous boat trips down at the harbour that the kids would enjoy, and you would be walking distance, I think, to the water park which I have heard is excellent. Hope this helps a bit.