Hey everyone! I arrived at this thread when searching for this problem and using Splashtop Business... so apparently this isn't something the LMI is the culprit for. I don't know what the solution is other than to make sure if you need to do remote access on a laptop that the display is active OR, in one case I had dual monitors connected via display port and then they were both turned off at night I had similar strange issues with the display and pointer accuracy. Hope this helps!
This strange behaviour seems to be tied to the "Microsoft Basic Display Adapter" that MS installs when there is no specific graphics driver installed (or Windows doesn't recognize the graphics adapter during installation).
On the local PC everything is fine, also when connecting via Microsoft-RDP or via LogMeIn with a local display attached.
Without local display the strange behaviour with LogMeIn (and only LogMeIn...) occurs, in addition you can't change display resolution via the LogMeIn symbol bar on top of the remote window.
Installed the latest Intel IGFX-Driver for the Intel P630 on the Fujitsu Primergy and everything works fine now.
I will check if this is the problem on the Win10 machines and come back and report.