Would you consider WWOOFing? You could actually do an entire month in Europe traveling around on less than $3k doing that - it's basically exchanging work on organic farms for a place to sleep, food, and a really fantastic experience. I traveled across the UK WWOOFing the summer after graduating from college, and it was great!
Another recommendation: use Google flights, flights.google.com, and click on the calendar button. Since it seems like you have flexibility in dates, this can really help you save money because the prices change a lot day by day.