If FSX is pausing when you bring up the FSHostClient chat window, you can fix that by going to Options / Settings / General (in FSX) and clearing the checkbox for "Pause on task switch." Unfortunately you'll still lose sound when the chat window has focus, but at least it won't pause the sim. There are also several options for how the chat window should work that you might want to check out in FSHostClient as well.
The jittery aircraft is a known problem I'm still working on. It's partly due to Microsoft issues, but I'm trying to work around them and see if I can make the remote aircraft more stable. Watch for updates to this in a future version of FSHostClient.
Aircraft sinking into the runway usually happens when you don't have the aircraft that the other person is flying, and the heights of the plane you're seeing the person in isn't exactly the same as the height of the plane he's actually flying. It's another difficult problem, but hopefully it doesn't happen too often with default aircraft.
Sorry I don't have better answers at the moment, but it's all still a work in progress, and hopefully Microsoft will release an update some time in the future that might help some of these issues.