So it can mean any summer or a specific one, can't it?
1) I plan to travel to Europe over the next few years. I plan to visit Greece in summer.
Does it mean any summer? Yes. To me, that means that at some point during the next few years when you are travelling around Europe, you will visit Greece. When you do so, it will be summer, but you don't know which summer yet.
2) - Do you have any plans for your holiday?
- I hope to go to Italy in summer.
It is "this summer". Is it OK without "the"? For me, this does not make it clear that you are going this summer. If someone asked me "Do you have any holiday plans?" and I wanted to be clear that I had plans to travel to Italy in the summer of the same year as the conversation was taking place, I would say "I'm hoping to go to Italy in the summer". If I was asked about general holiday plans I might say "I'm hoping to go to Italy this summer, Germany next February and South Africa next spring".