I'd like to copy selection shape in gimp and then paste it somewhere else (I've got first half of the image and I'd like do make it's mirror reflection). Do you know how can I do it?
You can use the "selection to path" button and then manipulate it as a path. Once your done with that, you can use "path to selection" to convert it back.
Do not use "selection to path" if you want to be accurate, paths are not intended for this kind of work. If you convert it to path and back while viewing closely, you'll see that it has changed its form a bit.
Selection is stored internally as a channel, so you can use Save to Channel in the Select menu to store it without losing accuracy. Then you can turn on the visibility of the newly created channel and work with it like with a normal layer. To restore, there is Channel to Selection in the channel's context menu and at the bottom of the Channels tab.
But what you actually need is Quick Mask, Alt+Q, small icon at the bottom left of the viewport. It handles all the internals of the described workflow for you. Turn on - and you can handle the selection like a layer, turn off - and the selection is restored. Internally, though, it's a temporary channel, you can switch to Channels and see it.
For simple cases you might also want to use transform operations on the selection right away, just switch Transform to Selection in the tool properties.