1. It must be a mobile remote control device that is able to initiate small tasks while moving on sloped or flat surfaces and negotiate valleys.
Definitely a large tri- or quadrocopter with either wheels or whats-the-name (a helicopters "landing gear") with high friction rubber - wheels may be a better option in some instances... I think.
If wheeled, the propellors can be run with enough power (and direction) to place and keep it where it needs to be. If needed, the propellors can even make it appear heavier, if you need that for digging out a shingle or whatever.
2. The main tool to consider is a quality camera (around 10 megapixels at least) that can preferably be seen from the ground on a laptop.
Only one I can think of... Canon makes a couple of DSLR's with WiFi, so you can frame the shots (and view them real time) on a remote (within reason) laptop.
Unfortunately, they ain't cheap.
Not sure how rain/water protected these models are, but a cover could be made very easy.
Other tools optional are a pitch gauge, and shingle gauge, and an arm that can use sidewalk chalk to circle damages.
Could you post photos of these gauges (with measures)?
As someone mentioned, spray paint might be easier - a cheap air brush and a CO2 bulb (as used on a soda syphon) with an electric valve for "air" would make a nice little marker "gun", but it would be more permanent than chalk of course.
If it has to be chalk, perhaps ground chalk could be blasted similarly to the paint or as a last resort, a piece of chalk could be held in a sort of flexible spring.
3. It must be able to stay on high slopes without trouble, so the wheel type is important. Ability to handle at least a 10 pitch roof is preferable.
That should be easy with a 'copter.
4. Flight is optional, or a mix of flight and ground mobility.
You said it.
5. Reasonable pricing is preferable.
That is the one parameter that is really really counterproductive! Sure, I know you haven't got unlimited funding, but anything short of the WiFi Canon DSLR will give you crappy shots that won't tell the whole story.
A 'copter isn't cheap to build either, as you need a pretty good lift to carry a DSLR, gauges, marker and the manipulators that's needed.
You're not constructing a toy where it's alright if it breaks down every now and then, but a pro tool that must work flawless at all times.
Unfortunately that costs a bit. Even if it was mass produced it would be expensive and you want to build a single unit - you sure wouldn't like the price if you asked an engineering firm to develop it.
Your best bet at keeping costs down, if you are up to it, will be to do all of the physical work yourself, obviously, but then you need to be at least fairly proficient in mechanics and be at least able to follow and understand an electronics schematic and finally program a microcontroller (or learn quickly).
Might sound like a hard journey, but the upside is that you get to select the materials - Carbon fiber, Kevlar, or Twaron fibers, titanium etc. is fairly reasonably priced if you just want raw stock.
6. Any way to view what the robot is doing from the ground is preferable.
And that's from a guy who prefers Pentax (who unfortunately doesn't make remote framing cams) and for some marginal stuff Casio.
7. It needs to take measurements of the roof
Photos with a reference object is a fairly accurate method.
8. It should withstand hot summer heat and icy cold winters.
For how long at a time?
I hope you get your own personal job security 'bot and don't be afraid to take the leap, one user here (Tommy) has gone from zero to a substantial do-"all" giant in a very short span, by expanding bit by bit (and next time he writes, I'm sure it's a guard dog as well
You can get lots of help here, but ultimately, it's a question of how well you can construct it, so I'd like to ask you about your previous results with tools and materials.