What dangers should I be aware of when doing a long-distance bike trip through Europe?

There are a lot of good answers already but one aspect has so far only been mentioned in comments: weather.

I would consider November a bad choice for such a trip and recommend late summer or late spring instead. Temperatures below freezing are definitely possible, especially at night. Snow is unlikely but cannot be excluded. More importantly rain is quite likey and most people would consider say 5°C and heavy rain as not ideal biking weather.

Additionally November has one of the shortest day light times of all months. So unless you only bike for a few hours per day you might have to add darkness to the 5°C and the heavy rain. This is not only unpleasant to bike but also drastically increases the risk for all kinds of road accidents.

This is more general travel advice, but complements the other bicycle specific answers:

I carry my passport in a thin cloth "wallet" with thin flexible cloth strap designed for use inside clothing. While most people wear this somewhere on their chest under their shirt, I arrange it to sit offcentre and under my belt. This can be arranged to be comfortable for sitting and walking. Cycling might make it harder. I also add my main large currency store (big notes) and VISA card. Paying mostly with cash allows me to keep the VISA card away from most random exploits.

Pickpockets are anywhere. They can be very good at pocket and backpack entry. I use zipped pockets wth the zip-pull removed. Bag zios can be secured with a cheap non-latching carabiner to greatly slow them down. Latching carabiners are more annoying for you – and for them. I’ve foiled would be bag-attackers on several occasions.

Cut resistant bags MAY be worthwhile for most valued objects. These are available commercially.

You can buy or make stainless steel "nets" to enclose bags. These make access hard and can be locked to a pole etc if wanted. Nothing beats a good bolt cutter. I wove my own net for an extended world tour. It was never tested by anyone, as far as I was aware :-).

GPS tracker in baggage may be worth its weight in lost bags.

Rabies: I was told by a travel doctor in Australia (while I was having my second of 4 Rabies shots after having been bitten by a dog in China) that Rabies takes time to progress from limbs to your central nervous system. Degree of wound and distance from core help give you time for a Rabies vaccine update to work well. So, yes, as Michael Harvey says, a bitten arm is far better than face. He said that substantial bites around the face are usually fatal in a few days.

Stinging insects, plants and other creatures are everywhere.
Carrying a cheap effecive treatment to counteract the venoms of insects, bees, marine stingers and stinging plants can be a life saver, or more often of great help. Sold under various brand names (STINGOSE in NZ and Australia) a 20% aluminium-sulphate solution works superbly not only on minor insect stings but on jellyfish stings and assault-grade stinging nettles (far worse than eg English ones). I avoided a hospital trip once after being stung severely by nettles over quite a lot of my body surface – Stingose worked very well – immense pain all over was reduced to still rather sore, but bearable.
Here is the 1980? NIH / Medical Journal of Australia announcing its discovery and how it works. A new and effective treatment for bites and stings

• Stingose, an aqueous solution of 20% aluminum sulphate and 1.1% surfactant, has been developed as a wide-acting and effective first-aid treatment designed to counteract the venoms of insects, bees, marine stingers and stinging plants. Rapid application to prevent further systemic abosorption is necessary. As virtually all characterized insect, marine, and spiked-fish venoms contain a protein component, it is proposed that the Al3+ ion in Stingose interacts with proteins and long-chain polysaccharide components to denature and inactivate the venoms.

one thing I did not see mentionned yet: watch out for landmines.

There are still a ton of them lying around in Serbia, Croatia, Bosnia and Herzegovina. So stay on the roads and do not walk around cluelessly in the woods. The touristic roads are safe, though.

I’ve cycled down the Danube to the Black Sea some 10 years ago. In my experience the biggest danger to your life are probably stray dogs, as mentioned by other answers.

Most of my luggage got stolen in Tulcea, Romania when I was camping close to the lake there. The thieves had the audacity to open my tent while I was sleeping, grabbed everything they could and escaped over a wall.

I also encountered a group of hostile adolescents in Romania, but fortunately they only surrounded me and made stupid jokes and never resorted to physical violence.

You can also expect beggars to be quite aggressive.

Keep your money, bike and luggage close. Try to make stuff look cheaper than it is. If somebody asks you how much your bike or bags or camera cost, make up some low number. Always keep some spare money on your body so you still have something in case everything else gets stolen.

Car traffic didn’t feel especially dangerous in Hungary, Serbia and Romania. Most drivers there are very used to cyclists and horse carriages. Don’t expect them to follow road laws though.

If you are really starting your journey in November, be ready for some cold and bad weather if you go to higher elevations. Close to the ocean it usually doesn’t go below freezing, but on mountain passes it can get dangerously cold pretty quickly. Snow and ice can make travel impossible (studded tyres would help, but are a nuisance on normal roads).

I have experience with long distance cycling in many of the same countries you will be going through (among others Romania, Hungary, Serbia, Croatia, France) and it surprised me as well, but I have the impression that car drivers in Eastern European countries are often much more considerate than their Western European counterparts. I am only speculating, but I guess they are more used to slow-moving vehicles intermingling with regular traffic. At least in Romania and Bulgaria, horse carriages are still common to see on regular roads and most car drivers are very aware that they must overtake with consideration and keep a distance.

I must admit that I tried to avoid larger roads and that most of the 600km I travelled through Romania were on rural roads with not so much traffic, but I also had a few short passages on larger roads with heavy traffic where no other options were feasible, e.g. around Turnu Severin, and I can’t remember having any larger issues there either.

Thieves and robbers may of course be a problem anywhere, hearsay about Eastern European countries is perhaps not the best in this regard, but you must just realize that you will have to take risks. I travel mostly alone and if you have a bicycle packed with luggage, luggage there are no means to properly secure, you just have to cope with the fact that many tasks are impossible, even simple things like going to the toilet for a number two, without simply leaving everything you have unattended and hope that it will still be there when you are finished whatever you are doing. That is life.

The only real problems I have had, and I must second Danubian Sailor here, were with dogs. Even quite often. Do not underestimate them. There are occasional stray dogs in the countryside and loose guard dogs in many villages and they are for some, to me not obvious, reason often triggered by cyclists. Expect them to set after you and go for your ankles. Using Google, you will find plenty of suggestions how to deal with the problem, and as a final resort you should be prepared to defend yourself or even hurt the dog as a mean to show that you are stronger.

What has worked for me, but your mileage may vary (also my mileage in the future) is to:

• Either outrun them – a dogs stamina is usually not very great or perhaps they decide that the effort to keep up with you is not worth whatever gain they expect from catching you. You should though expect that the dogs can easily reach 25-30 km/h and keep up with you a kilometre or more. That is indeed faster than my ‘comfort speed’ when cycling, but with a potentially rabies infected biting machine literally on or in my heels, that has usually worked out.

• Or, if you realize that you can’t outrun them, stop, get off the bicycle and place the bicycle between you and the dogs. Notice that you have to decide for this option with enough distance between yourself and the dogs for you to have time to position the bicycle as a defensive barrier. I think I’ve only used this option twice and the outcome was very different. The first time was uphill somewhere in the Greek countryside, where I had a single German shepherd lookalike turning up from nowhere and going after me. As soon as I got off the bike, the dog started wagging its tail and turned into a very friendly fellow. As soon as I tried to get onto the bike again, the dog started to snarl and show aggressive behaviour. Don’t ask me why it worked, but I solved the problem by walking the bike for a while with the dog following and after a while he lost interest and ran back. The second time was in a village in Romania with a broken gear cable, where I realized that it would not even make sense to try to outrun the two dogs setting after me. Here, it did not help to get off the bike, but the the dogs actually staid on the opposite side of the bike and did not have the guts to go around the bike to attack. I could probaly have solved the situation with pepper spray, which I did not have. I was ready to use the bike lock as a weapon if the dogs really tried to attack, but after a while they lost interest and went back to their home.

Just a word on rabies, in case someone didn’t catch my slightly humorous tone in my statement about ‘rabies infected biting machines’: Rabies is in reality nothing to worry about in the countries OP is going to traverse. There have in total, in all countries along the route, been five cases of human rabies in the last 10 years (one each in Romania, Italy, France, Spain and Portugal). If OP is concerned about rabies anyway, he should get a vaccine in advance and not rely on being able to be bitten in a limb far enough away from the central nervous system that he has time for a post-bite vaccine before the infection reaches an irreversible and fatal stage.

FWIW, I met a bear out in the wild in Romania last year, we were probably not more than 4-5 meter apart, and felt much more in control than with these two dogs. The bear encounter was more like, ups, neither of us ment to disturb the other, if we just forget that we met, we can both go on doing what we were just doing. She was friendly enough though to sit down for a photo before we parted.

You have pointed out that most common dangers you’re going to meet in the eastern part of your journey, with one significant exception: it’s not wild animals you should be afraid of; the most dangerous animal you’re going to meet is a dog.

Dogs, especially in packs, are a really big danger especially to travelers moving unarmored with big velocity: that’s you. Dogs in villages are kept on chains to protect houses, so the aggressive ones are preferred, but they manage to escape from time to time. They may join other ‘refugees’, and they present a deadly danger you should under no circumstances underestimate. Most of my cycling colleagues from Poland have experienced at least one attack by a dog.

Thieves are a menace mostly in cities, and robbers on the main streets. On the country roads, you’re less likely to meet them, however, you’re more likely to meet dogs.

Aggressive drivers, yes, but they’re unlikely to try to kill you for fun. It’s the stupidity that makes them murderers. They are annoyed driving behind you kilometers long and they try to overtake aggressively despite having no place. Still, it is unlikely to be a nuisance on local roads, with little traffic, and there’s unlikely to be any issue to overtake a cyclist. And people driving through villages are more used to cyclists than those from cities – bicycles are a very common mode of transport in rural parts of Eastern Europe. Most older people don’t have a driving licence, and cars are too expensive if you don’t have a well-paid job.

You should also be aware of hooligans and drunk young men going to or coming back from parties. They may be aggressive and annoyed by your presence.

In western countries, cycling for fun is popular and relatively safe. You should be careful riding on main roads, but everything else is more civilized.