I haven't heard anything about Laszlo testing Adobe AIR apps, especially with the AIR releases in the past two years. The AIR browser has never been supported officially.
I'm assuming that you are still embedding an SWF into a HTML page for your AIR app. When using AIR with OpenLaszlo, I'd always generate an ActionScript 3 based AIR app (not embedding an app SOLO compiled to an SWF into an HTML container). That way, I can use Flash Builder to debug any problems, which makes things a lot easier than debugging embedded SWFs. If you plan to deploy a DHTML runtime app, there's of course no way around embedding the OpenLaszlo app into an HTML container.
Update: Flash 10.2 and higher support native cursor integration (native here means operating system), which provides MUCH better performance. Here's a general description of the API in an Adobe blog post: http://www.adobe.com/devnet/flashplayer/articles/native-mouse-cursors.html
It's technically possible to use the native cursor with OpenLaszlo, I've created a proof-of-concept - but it only works with a heavily modified version of the LzMouseKernel.as class (from the SWF9 kernel files in the LFC), since the kernel will always reset the cursor for various mouse events. And since native cursors require at least Flash 10.2, this is only possible with the SWF11 branch of OpenLaszlo.
I suspect that the problems you are seeing are connected to the way OpenLaszlo enables custom cursors (which is done by hiding the native mouse cursor and moving an sprite around following the invisible mouse cursor position).