I've never flown to Europe with a bicycle, but I have rented a bicycle in Europe several times. It's pretty easy to do, and most big cities have a number of reasonably-priced options. You can usually rent a bike with racks and panniers if you want.
Airline bicycle policies change periodically. It used to be inexpensive, then it got crazy expensive, and it's currently rotated back to inexpensive again. Most airlines have liberal size limits for bicycles (and don't even charge you the oversize fee), but a few still have size restrictions that make it practically impossible to take a bicycle because the maximum size of the luggage (at any price) isn't big enough for anything but a folding bicycle. Also, it is not possible to take a e-bike on any flight on any airline (because of the battery). It's important not to book your flight until you fully explore the airlines bicycle policies. Even then, be sure to make a copy of the policy on the day you book, so that you can show it to the agent when you check in, just in case the agent thinks the policy is something else. I just checked a few airline policies now for flights from the USA to Europe. All of these airlines have generous size limits for bicycle boxes.
- United Airlines: A bicycle just counts as any other piece of luggage, except that oversize fees are waived. I've taken my bicycle domestically on United many times with no problems. United doesn't even have a way for you to tell them that you're bringing a bicycle, so just show up with it. But pay attention to the difference between "Basic Economy" and "Standard Economy" and be sure to book the one that fits your needs (e.g., Basic Economy does not allow a carry-on).
- British Airways: A bicycle also just counts as any other piece of luggage, but the oversize fees apply. You have to tell them in advance if you are bringing a bicycle.
- Icelandair: A bicycle counts as a piece of luggage, plus a hefty "odd-sized item" fee will apply (as much as $210). You must reserve space for your bicycle in advance. Icelandair still tells you to deflate your tires--poppycock! In my opinion, Icelandair's policy rules them out. All the other airlines are better.
- American Airlines: A bicycle counts as a piece of luggage. Oversize fees are waived, but if you are flying to Madrid, the size restrictions will make flying with a bicycle impossible. If you pack it in a cardboard box, they will make you sign a damage waiver.
To smooth the passage through security, and to avoid overweight fees, it is always advisable to pack
only your bicycle in the box. Do not include any of your gear. To avoid overweight fees, I always pack my saddle, pedals, and front rack in my other bag. This gets my box down to 47 pounds (the limit is usually 50 pounds). Weigh your loaded bike box to find out if this is necessary for you. If you put only your bicycle in the box, security will just open the box, look inside, and probably do an explosive residue test. If you cram the box full, they will probably have to unpack it, and they almost certainly won't repack it as well as you did.
You usually have to take your bicycle to an oversize luggage area, so you should arrive at the airport plenty early. I'd recommend three hours for an international flight. It might take some time to navigate the special process.
It used to be that all airlines said you had to deflate your tires. These days, most (but not all) airlines have removed this idiotic rule. There's no chance that your tires will explode as they fear. Even if they tell you to deflate your tires, I would ignore this rule. I might take a little air out so that they are a bit soft, but I would not fully deflate them. Inflated tires offer significant cushioning for your wheels and bike. Keeping air in your tires also simplifies your chores upon arriving, as you don't have to pump them back up.
Try not to use multiple airlines for a single trip. That just complicates matters and increases the risk of damage. In fact, try to book a single, non-stop flight if possible.