Europe is generally very bike friendly - from trains, airlines, cycling network, storage, parking etc. Some countries, like Denmark, Belgium etc lead the World in terms of cycling and infrastructure.
I flew home from Belgium to Portugal yesterday and there was a Pro woman with her bike - in an EVOC hard case - and someone with a foldable bike, zero protection, attached to a large suitcase. Both looked perfectly happy when they received their bikes at the other end at the oversized luggage area.
There are a number of regular visitors to where I live who come here for training in the great weather and they all bring their own bikes, never an issue.
I've also noticed that Pro cyclist Willie Smit just uses a cardboard bike box for his very regular trips around Europe for races - he shows this on his Youtube channel.
Re renting, I couldn't get a rental on this last trip, despite the Classics race season being over, all booked - in the end, I used a borrowed bike from my brother-in-law but otherwise would have taken my own. The cost for flying a bike in Europe is around 50-60euros per flight which is similar to bike rental. Next trip, I'm taking my own.
Re buying, it is a very nice idea, especially if it is a Colnago where they do have new owner collection options with a lot of fanfare.
