I am not with Laminar Research; just a flight simmer.
By using the demo mode I assume you have downloaded the digital version. There is no difference, as indicated in other locations in this forum, between the demo version and the "full" version other than making a purchase and then entering the "password/licence key" provided at the time of sale/purchase. The only restriction is a time limit on use and airport location. So the demo vs full version theory can be eliminated.
Because of the way you indicated you want to use the software I would suggest you contact Laminar Research at email@example.com to ascertain if you should have a professional licence instead of the "single user" or "home use" licence.
In relation to the "melt down" or crash after running for 15-20 minutes it may be that your laptop is overheating. X-Plane is extremely resource hungry and generates a lot of heat. More so with the XP11 version. To reduce the impact a good cooling system needs to be built into your machine. Although your specs indicate (by Dr Google) it is a laptop gaming machine there are some machines better than others for X-Plane.
There may be other factors causing the problem. To ascertain if there is, locate and attach a copy of the log.txt file, which is generated immediately after X-plane is shut down, as part of your question. In your case when you have your next "melt down". This will allow others to view and try and ascertain a possible cause.
How do you attach a file? Have a look at the following link in this forum at http://questions.x-plane.com/16206/how-to-insert-a-file-in-a-question
Good luck in solving your problem.