Honestly, I haven't seen the oval sticker since we started putting country identifiers on our license plates.

I once asked the Estonian vehicle registration agency about this, and their answer was that they are happy with a non-EU-plated tourist vehicle staying in the country for 12 months. Other countries might give you a different answer.

Yes, if you take the car out of the EU/EEC, the car's temporary import permit resets. (Be mindful that some previously convenient visa run borders are currently not viable, such as all of Russia, Belarus and Ukraine. Serbia and Moldova should count though, as will Turkey or Morocco.)

Also be mindful that even if your vehicle can stay for 6 or 12 months, you personally are only allowed to be in the EU/EEC/Schengen for *90 days out of every 183* consecutively, so doing a visa run on the 89th day and immediately coming back won't make *you* legal.

Finally, consider the implications that there are several different overlapping legal regimes that are relevant to you:

* The European Union;
* The European Economic Area, which is the EU plus Norway, Iceland and small places like Liechtenstein or Andorra - no border checks;
* Switzerland, which is not part of either but has enough treaties and equivalent laws to feel like it is - the border post will probably just wave you through unless you're a cargo vehicle;
* The Schengen area, which is most of the EEA and Switzerland, but not e.g. Romania or Bulgaria or Croatia or Ireland - have to show ID at the border;
* The UK, which is not a part of any of the above, even if most Europeans don't need a visa for short visits.