The fees and terms apply for each individual bank and for that reasons are not the same for all of the. In regards to the Debit card then all of them offer a debit card.
There are several banks to work with i.e. USB Bank Cyprus, Marfin Laiki Bank, Bank of Piraeus, Bank of Cyprus etc. you may check them out on your own.
You may also have a look at the Caribbean bank account that we offer here on the forum; /forum/f5/caribbean-offshore-bank-account-100-anonymous-banking-secrecy-18458/ Maybe you can open one of each.
Thank you so much for your help I will go through the banks to see what will reflect my needs. So I can open a bank account both in Cyprus and Caribbean, can I open one somewhere else also?