The cost of a Windows 7 Home Premium or Windows 8/8.1 Core license is less than the added cost of getting a small-production Linux machine, in general.

Plus, with Windows 7 you can always reuse the key to run a VM or on another machine (with the key embedded in ROM on most Windows 8 systems, it's less clear how practical that is to transfer.)

Pushing your budget, but I bet you'll like this one:
https://www.system76.com/laptops/model/galu1

I have no idea how easy those are to open, so you may want to look into that before deciding whether to get it with SSD preinstalled vs. upgrading after purchase. Their prices on SSD upgrades are relatively good.

I'd avoid dual-core i7s, they're not worth it. If you need quad-core, you have to have an i7.

Second, do you need a full-speed processor, or (on the i5s) is an ultrabook-class processor OK?

Fair enough.

IME, it depends on what you do with it, and which desktop you run. Running KDE day to day for work, I find it uses about the same as a decent install of Windows, and am more comfortable with 8gb for a general-but-heavy use.

(Software development on my work desktop uses a lot more, of course.)

Should be doable at FHD; "or better" is unlikely without pushing your budget.

How thin/light?
14" FHD is pretty unusual -- it's actually more common at both 13" and 15"

Should be pretty routine these days, although except for in very thin/light models it's pretty easy to swap a HDD for an SSD.

Also, if you mainly care about speed and not portability/being drop safe, you can with many models buy it with the cheap HDD and just add an mSATA card to have both.

Why do you want discrete video?

BTW Nvidia with the open-source (Nouveau) drivers plays nicer with Linux than ATI does these days, and that's not all that well... and if you want switchable graphics, both are problematic but I've had pretty good luck with Bumblebee (a Linux implementation of Optimus.)

Meanwhile, the Intel graphic drivers for Linux are well-supported and nearly bulletproof, and on the new ones, quite fast.

If you care about graphics performance, drop the 3rd-gen processors and just look at 4th-generation/Haswell -- the HD5200 ("Iris Pro") models are not really common, but they will outperform the Nvidia or ATI discrete graphics in most 3rd-generation models, including some lower-end gaming and workstation models. The HD4600 models will perform at least as well as lower-end business graphics from ATI or Nvidia in the 3rd-generation systems (e.g. a Geforce 630M or NVS 5200M)

If you really need performance beyond that, you're going to have trouble finding something reasonably light in your price range, and be looking at a relatively beefy GPU. The Lenovo Ideapad Y510p is heavy, but otherwise may be what you're looking for: http://shop.lenovo.com/us/en/laptops...-series/y510p/ (It IS an Nvidia-based system, though.)

For a lighter machine at a higher price point (although I'd rather have the Galago, and the Iris Pro at the same cost), look at the Lenovo Thinkpad T440p.

If you want something lighter, and don't need the discrete graphics, consider the Dell Latitude E7440 (aka "Latitude 14 7000 Series Ultrabook". It's at the high end of your price range, but if you wait for one of the regular 25% off coupons for the Dell Outlet (via http://twitter.com/delloutlet ) and don't mind refurb, you can get one under $1000, and they're a very light machine with a reasonable amount of power. (I can't figure out how to order a new one with the 1920x1080 FHD screen via the public web site, so if you wanted to buy new you'll have to deal with phone sales.)