I'm currently trying to pick between the 3 also, with the current Dell offers, its making waiting to see how much the razer will be here in the UK quite hard!. The XPS 13 with the slightly better i7 and the iris 540 in benchmarks runs a bit better than the stealth (Looking at the benchmarks from notebookcheck), and looks like battery life will still be a bit better on the XPS 13 from reviews.
The XPS 15 is tempting, mainly the 2 sodimm slots rather than the soldered ram in the stealh and xps 13, but i think its a bit out of my budget. In terms of the performance boost you will get with the 16gb ram spec on the 15, for only around £100 more is a steal . Especially as the reviews i've seen mention it is still quite lightweight, so shouldn't be too cumbersome to carry around. The main difference would be with the more mid-range 15 within your budget, would be the 1080 non-touchscreen. The quad core xps 15 will likely be a lot better if your going to be running programs like matlab/stats modelling for physicsy type stuff if you use it.
My usage is very similar to what your looking for, a lot of pdf annotation/document editing. a few games of CS:GO and general web browsing. I'm probably gonna wait for the stealth pricing, and hopefully dell will have some more offers soon if i still want to go for an XPS.