Hi, Christian! You'll find that every
answer here starts with "it depends," especially if you don't say where you're located!
In my experience in eastern Kansas, strawberries don't need frost protection unless they have actual berries growing when the frost threatens. On the plant-side of the placental barrier, they have antifreeze in their veins, so you can stop worrying about covering them. And they will happily grow among tree
roots and other plants.
What you do need to worry about is how to identify the older plants, as they will stop producing berries after 3-4 years. You will need some kind of system to tell one year's plants from another's, or else after a few years you'll just have groundcover and no berries. The "traditional" (maybe a couple generations old) way to do this is to plant in tidy rows, so that you can just pull up or till up the row containing last year's plants, leaving this year's on either side. Another approach is to build a tiered planter so that the younger plants are generally lower down than their parents. Neither of these approaches lends itself to planting under trees.
I've tried using different colors of soda
straw or plastic
spoons, only to find that the plastic degrades in sunlight and is gone by the time I need to find it. Consider different colored landscaping rocks or something else really durable. Otherwise you will be facing the day when you have to pull up ALL the plants and start over.
Good luck!