I tried to meet your needs with a project that has as main objective to minimize the interventions on the existing villa.
Your house is very beautiful and I don't think it's a good idea to ruin the existing balance by creating new volumes or heavily modifying the existing one.
The most important changes in my project concern the interiors so that I can meet all your requests.
Outside I simply created an outdoor area with dining and living room (West side) and an outdoor area with hot tub on the North-West side. There are also other focused interventions on the facades (creation of new windows or closing of existing openings) which, in my opinion, fit perfectly into the overall design.