Personally, I'd go with option 1 as desktop end up being cheaper on the long term since you can only change certain pieces to keep it up to date and able to run most recent games on high graphics.
Though, I think you should consider the usage you make of your computer before anything else, I always wanted a desktop though it took about 5 years before I finally bought one after getting into gaming PC. Like you I moved around sometime and I liked being able to bring my laptop with me.
I had two gaming laptop, the first one was an Alienware which I had for 3 years, it was great! Probably the best laptop I've ever had, no overheating.. I changed it because the battery was pretty much dead after 3 years, I was using it a lot for school too. The only downside was the price, it was rather expensive, over 2k, and that was 6 years ago, these days Alienware are even more expensive, specially if want a 17inch screen like me on your laptop.
My second laptop, which is the worse thing I ever bought in my entire life was an Asus Rog I think... keeps overheating all the time which makes playing certain game a pain with the throttling and frame drop that comes with it.
There are a lot of good gaming laptop out there, just make sure you buy one that's big enough to have a good airflow and fans, as you overheating really is annoying and it'll happens even if you play a game like neverwinter night.
If you buy a desktop, I really don't suggest Alienware, while they're not bad computer, they're extremely expensive compared to other desktop you'll find if you look around. If you have the money though, well I was very satisfied with my Alienware laptop, I'm sure the desktop version as just as good.
If you're feeling confident, you could always build your own custom desktop... which is what I did personally and its great! Though it can be a bit hard if you have never done it and there is always a risk to break a pieces while manipulating them.
Overall, if you move around a lot, I really suggest you get a gaming laptop, there are good gaming laptop out there, sure you'll have to spend a bit more money on it then you would on a desktop, but I personally never regretted buying laptops during the time I was moving around a lot for school and stuff.
All of that is based on my personal experience, its not much.. but hopefully it helps ^^