Smart phones tend to have sucky battery life -- if you're running a GPS app, they'll generally be dead in a few hours. They also don't have ANT+ compatibility that I'm aware of. But external power sources are an option. Get a basic Android pay as you go phone (do make sure it has a GPS, though most do), don't actually pay for service, install osmaps (that's the name of the package), preload local maps ... you're good, for a few hours, till the battery dies.

Car GPSs would work great, and they're cheap as well (good ones can be had for under $100, just make a mount of some sort) -- but they tend to have even worse battery life, being designed to work with 12v power available. Which is a pity, but of course they could be given external power sources as well. No cycling specific features, however.

The Garmin Edge 6**, 7** and 8** models cover everything you asked for except for "being able to make routes without a computer" (and I'm not sure what does cover that, though if you just want it to take you to an address or point of interest, they can do that without a computer) and "being affordable" (though the 605 and 705 are findable used relatively inexpensively.)

Linux compatibility depends on what exactly you want to do, but I can access my 305 (which has no navigation, so you probably don't want one, just to be clear) and 705 (has navigation) and read my history from them via Linux without too much grief. (The 705 is easier, as it's seen as a mass storage device, so you can copy files directly.)

My Edge 705 is very slow at calculating routes -- you tell it where you want to go, and it'll take a minute or two to find a route if it's across town. And it has to recalculate when you get off the route it's assigned. But it does work.

It does not have a touch screen, so it works fine with gloves. The 805 has a touch screen, so you'll need to take the gloves off or use one of those pens, or just get gloves that work with capacitive touch screens (they do exist.)

The Garmin Edge devices are water resistant -- rain will not hurt them, even a downpour. They say they'll survive 30 minutes of complete immersion in one meter of water -- but I'm not testing that. But even hard rain, for hours? No problem.