We want to travel Europe for a year or so in a car we buy over there. Can we do that and what about insurance?
In France, it is possible for tourists to lease a "transit temporaire" vehicle. This is a scheme whereby a new car for use by non-residents can be registered tax free, and is then rented to for a temporary period, deferring the taxes. At the end of the period, the manufacturer gets the car back, and pays the taxes on the (now depreciated) vehicle. Due to the tax advantages, you're unlikely to find a cheaper way to rent a car, and this is much easier to organise than buying a car. The car receives an attractive pair of red registration plates.
The length of lease possible seems to vary based on your situation. Insurance is included. The car can be delivered to locations outside France if required. Since the service is explicitly targeted at non-residents, you must prove that you aren't resident in the EU in order to qualify.
In order to compare the different schemes available you should search for "europe tax free lease".
Europe is still 27/28 different countries with as many different legislations. So it will be hard to answer your question in general. For each single country you may get answers here.
So let me start with Germany:
Of course, you can buy a car, but obviously you will have to insure and register it. In Germany, this is going to be the sequence of things, i.e. you first have to buy insurance, then you go and get the car registered. In order to register a car in Germany, you will have to be a resident, which I guess you will no be able to produce.
That means either you have any nice person (relatives?) living in Germany which will register the car on your behalf or you will be out of luck.
Please note that AFAIK the UK (part of the EU until March 2019) does not have that concept of registered address, so it may be easier there. Also of course as you can do everything in English there.
Just be aware that the UK is one of the very few countries in Europe which drives on the left. It may turn out very inconvenient if you buy a right hand steering car in the UK and then spend most of your time in continental Europe where we drive on the "right" side of the street.