Good morning Yvonne,
Thanks for your queries! I can completely understand wanting to prioritise some aspects of your upcoming trip. With regards to timings, this would really depend on how much time you are planning to spend in each area you are planning to visit.
As you are flying into Dublin, it could indeed be an option to explore Dublin whilst you are there and then travel across to Donegal and work your way around the coast towards the Antrim Coast and Giant's Causeway and down towards Dublin again, or perhaps as you've mentioned yourself, heading towards Donegal and exploring there first and leaving Dublin to explore near the end of your trip. Two weeks would definitely be a realistic time frame for this.
A lot of visitors to the island tend to book accommodation as they travel, so I don't feel that this would be an issue, however, I do usually recommend having a browse of potential accommodation options and perhaps reaching out to them beforehand to get an idea of what their availability is like during your travel dates.
You may find our Traveller Health section
helpful when it comes to healthcare options and insurance queries. As you've mentioned, the European Health Insurance Card does indeed cover most medical care, although it would be recommended to query any specifics with your travel insurance company.
I hope this information helps a little bit further with regards to your plans. I hope that both you and David have a wonderful trip and if there is absolutely anything else that I can provide guidance with, please let me know.
All the best,