There are a lot of reasons for this to fail from the TV just doesn't do this to a firewall in the router.
But I see something off in your post. That is, the TV remote should not do a thing. That is, I know of a LG with a mouse remote and the mouse cursor will show but nothing should happen on the PC side. That's expected.
The no desktop icons may be telling us the PC is extending the desktop. I can't know your level of Windows expertise so I may not offer enough or too much. So try moving the PC mouse off the screen left, right, up and down and see if it shows on the TV.
Miracast is usually a dead end for most as you try to play a game or video there.
I am trying to use Miracast for my LG TV. I was actually able to connect it through the screen share feature. However, all I am getting is my desktop wallpaper with the task bar below. The shortcuts are nowhere to be seen and I am unable to open any windows. It's just a static image. I am able to move my mouse (or the remote operating as the mouse) but it does nothing when I click on things.
Oddly, it opens windows on my actual PC, but not in the TV itself.
Any idea how to make the TV display my PC screen in any usable way before I give up and just hook it via HDMI?
I have Windows 10, if it matters.