That's correct.

To OP, you can sign into your own Paypal account and click on the "Send Money" tab. All the info required for Paypal transaction is there. I assume your villa (rental?) is not an auction transaction. That's why you will use "Send Money" when doing Paypal.

I believe Paypal imposes less restrictions opon buyers/renters when Personal Paypal accounts are involved. I see no reason why you can't fund your Paypal Personal account via credit card. You will have to go through the Paypal verification before you can draw funds from your cc though.
That's not true.

Even though I haven't sent any non-US currency payment using Paypal, when I click on "Send Money' tab, there are quite a few currencies options other than USD. I assume it means you can send non-US currencies as well.

I know for a fact Paypal charges buyers both transaction and conversion fees when the payment is non-local currency. What's not clear to me is who will charge you, the buyer/renter, for the currency conversion fee if the Paypal payment is funded by a credit card. I assume the credit card company will ding you with the 1%-3% currency conversion fees. If Paypal allows you to specify and send non-local currency payment, will they double-dip and charge you for the conversion fee as well?

Outgoing international wire can be expensive. Do shop around and see if Paypal (or wire) is the better deal.

Nevertheless, there is currency exchange/conversion involved. Make sure you pick the card that gives you the best rate. Visit this thread for more detailed discussion: Best card for foreign exchange?

Another idea to earn you more points from Amex. Ask the owner if s/he would take Amex Traveller's checks. If yes and you happen to live close to the Amex Travel Offices, following the instructions in this thread, you can pocket quite a few points effortlessly. @:-)