Check it! The trees outside do fine through the winter, just because it is small does not mean it is not a real tree. Its fully equipt to handle the cold! You can winterize your tree, and just make sure you water much less than you normally would if its below freezing as you dont want the root system to freeze. The dirt should be harder in the winter due to keeping it dryer, and if it gets very very cold bring it in (not sure what winter temps are like in the UK)
Hmm. I have never heard of these wild strawberries. However, strawberry plants seem comparatively harmless and unintrusive, unlike say, blackberry plants, which are hard and thorny.