If you purchased Lion for your desktop, then yes, you can upgrade your laptop without having to purchase Lion again (assuming of course that your laptop will run Lion). Just log into the Mac App Store from your laptop using the Apple ID and password through which you purchased Lion and you will be able to download and install it without extra charge.
As to application compatibility with Mountain Lion, that is definitely an issue, as it is with any major (and sometimes minor) operating system upgrade. Here's a useful compendium:
http://roaringapps.com/apps:table
though you should contact the developer of any apps that are critical to your use to get the official word.
Regards.