I see what you mean about the gap between concrete and fence and the potential for weeds to creep through from neighboring gardens. I’m constantly in battle with next doors ivy:(. So yes I think raising the borders would be a good idea but as Sonia has pointed out that trees and shrubs will be far better planted in the ground rather than in a planter. I think a raised bed would be an ideal solution. The only issue is that any soil which is next wood will cause it to rot and create more problems for the fence, easily solved by lining the wood with some heavy duty plastic sheet. Just cut a section so it covers the wood and able to let water drain in the ground, then at the front edge of the border (next to the grass) you could use whatever decorative edging you like. Once it’s filled in with soil/compost this will cover the concrete and give a nice uniform look. If you want to use weed proof membrane and pebbles then that’s fine but cut a hole for any trees/shrubs so the roots can use the ground soil. Also, if you use bark chips as a top layer this makes it much easier to pull out any airborne weeds such as dandelions. Here’s some inspiration (notice on the second pic the plastic backing next to fence). Hope this helps a little:)
Q
Rustic feel
Q