If you say where exactly in Cyprus you plan to stay we might be more of a help. In Larnaca Palm Beach and Golden Bay have heated indoor pool. For Limassol Twig...
There are many car hire companies, mainly local that allow cars to cross over but with extra 3rd party only insurance. A new crossing point has just opened in D...
If you drive it is 45 min by straight motorway ride
You will need to hire a car and stay in either Larnaca or Limassol. Seeing again your interests think Limassol might suit better for hiking on mountain villages...
Castle should be closed and I am not aware of a natural history museum in town. All restaurants will be open but I would suggest dinner in Sun Hall hotel
Not forbitten to swim but warned that on some beaches in that area there are strong currents making swimming dangerous, so warning signs are posted.
When do you plan to visit?
Try Travelexpress shared mini bus service http://www.travelexpress.com.cy/en-us/fares
Latchi is a summer destination and March is def not summer in Cyprus. If what you look for is a relaxing time and walks to Akamas it will be just fine
Pets move freely between EU countries provided you show vet records