Here's some information based on my experience of keeping a Canadian-plated motorcycle in Europe for the past 15 years:

1) Shipping - in the old days (pre-COVID), the easiest and cheapest way of getting a motorcycle from North America to Europe was to ship it by air, generally out of Toronto or Montreal to any continental European city such as Paris, Frankfurt, etc.

Total cost of doing that (one-way), all expenses in, averaged about $1,300 USD. That includes port charges at each end, documentation charges, etc. The moto would be shipped standing up inside a container, all you have to do is show up with less than 1/4 of a tank of gas. No need to disconnect the battery, crate it, etc... you could ride it off from the destination about 3 hours after the plane landed.

Nowadays, reductions in flight schedules have made air freight more expensive or perhaps not even offered. Contact a company that specializes in this, such as Motorcycle Express in New York state, to get up to date information on the situation. Also investigate Canadian airlines such as WestJet, Air Transit, and Air Canada, all of whom have offered motorcycle shipping programs in the past.

Shipping by air out of the USA is a pain because of all the fear, uncertainty, and doubt associated with travel by air in the USA (e.g. 'Take your shoes off').

Shipping by sea might look cheaper, but the port charges at each end are much higher, and the administrative overhead for you is a lot higher, not to mention the cost of getting to the destination port and recovering the motorcycle, un-crating it, re-assembling it, etc.

Don't rule out the possibility of using a household moving company that specializes in household moves North America - Europe.

2) Leaving it There - There are two aspects to this, doing it legally and doing it "not so legally" (but successfully).

Legally, all western & central European countries allow a tourist vehicle plated in North America to be imported for legitimate tourism (means, only operated by the tourist, tourist remains with vehicle and exports vehicle when he/she leaves country). Most countries allow 6 months for the vehicle, although this might be shortened if the individual (the human) is only allowed a shorter period, for example the Schengen region's 90 day limit on visa-less tourism.

So, the "rule" is you come in with your vehicle (or at about the same time, if you ship it by air or sea) and you take your vehicle out when you leave. That keeps you totally legal. Just buy liability insurance for it, and don't overstay your own (human) entry period.

On a more practical level (less legal) - no western or central European countries that I know of record the entry of a tourist vehicle nor do they require any "check-out" procedure when the vehicle leaves (the situation is very different once you get into Turkey, North Africa, or the far east, in general once you get out of the EC area).

So this lack of recording of vehicle entry means that as long as you are discreet, you can pretty much leave the vehicle in Europe forever. You just have to be discreet about it, for example:

• Never allow anyone other than yourself to operate the vehicle. There are dreadful taxation consequences if someone else is found operating your "tourist" vehicle.
  
  
• Don't ride around in the same town for long periods of time, the local cops will notice the strange licence plate and become interested. In other words, don't spend more than a week or so in any one place.
  
  
• Likewise, avoid showing up in the same town at the same place (friend's house) year after year.
  
  
• When you go back home or back to work, store the bike somewhere that it will not be noticed - at a friend's place, with the licence plate against the wall, or removed, or at a moto dealership that offers long-term storage.
  
  
• Be sure to keep your insurance document (green card), driver licence, passport, ownership, etc. with you when you ride. If you do get stopped in a routine traffic stop, you want the process to be simple and easy and complete.

So far as crossing borders is concerned, within the EC, it's a non-issue, there is no border control except in rare and exceptional circumstances, it's like crossing a state line. Into other countries that are not EU members, all I have ever been asked to show is proof of ownership and proof of insurance. Again, once you get into North Africa or anything east of Turkey, things get much more complex. Be aware that the suffix '-stan' on the end of a country name means "fecked up country, don't go there".

I hope this information is of use to you.