Why is print successful in communicating an idea/concept? Print in a traditional sense (litho) is very tactile; it enhances the users experience with different materials (touch) and can also achieve added dimensions (uv, embossing, die cutting, paper engineering, etc) that cannot be achieved with web/screen work. Print also allows you to use these extra dimensions to challenge the user and potentially make the message stay with them that little bit longer.
Why do you use print design as a communication tool? I think I've answered this above.
Considering your own experiences of designing for print; what do you consider to be the best/most enjoyable thing about the print design process? Again, I think this is answered above; design should be attractive and fun, but I think the best design is more than this - it's powerful, thought provoking, intelligent and clever - all of which print does differently (and some would argue) better than digital media.
Do you think that digital media or screen-based design will ever fully replace the printed medium? No. Take music for example, we (all the guys here) like having that very tactile experience of buying a cd, reading the booklet, feeling the material the pack is made from, etc. It makes the whole purchase an experience. I don't think you get this when you buy online. We all have iPods and love the format, but think we'll all continue to purchase something physical.
Why and for who is print design successful? (You may specify clients) potentially anyone! Taking two examples, WaterAid (charity) uses print to get a message to a wide audience in developing countries where they don't have access to digital media. But, very differently, Caffe Nero use print to promote drinks/food within their stores.
Do you design for print and screen? Are there cross overs, and if so how do they work? Yes. Both have their strengths and weaknesses. The best campaigns realise this and intergrate the two.