You're correct about the difference in sensor size. 35mm lenses will have a larger coverage area, so you're fine using them on smaller sensors.
Canon's nomenclature for APS-C, crop sensor, digital, DX, whatever everyone else calls it, is EF-S. These lenses are specifically designed to only cover the smaller sensor.
You can use an EF lens on an EF-S camera, but you cannot use an EF-S lens on an EF camera. The only
concern I might have is if the Sigma would work with AF and metering. Because Sigma, Tamron, Tokina, etc, basically reverse engineer the pin-out on the lens, and the communication protocol back to the camera, there is a very slight possibility that there would be a hiccup. I've got a Tamron lens that's 14 years old that I used on a Nikon film body, as well as 2 nikon digital bodies, and never had any trouble. However, caveat emptor. I would swing by a local camera store and take it in to make 100% sure. Any good store should let you take a memory card and your lenses and give the camera a test run out in the parking lot.
The consequence / benefit of having the smaller sensor, is that your effective focal length is magnified. Since you're projecting the same size image, but taking the middle area covered by a smaller sensor, you'll get a smaller field of view equivalent to a longer focal length. Your aperture ratings will be equivalent.
The other benefit of this, is that, typically with inexpensive lenses, the area of image quality that is the best is the center. By using a crop sensor, you're getting the best part of the lens, even with inexpensive ones. That 35-80 kit lens may look better than you think.
With the sensor Canon uses, you'll get a 1.6x multiplier on the 35mm length to come up with your effective focal length. So, given that, your existing lenses would all be pushed 1.6x more towards the telephoto range.
Sigma EX 17-35mm F2.8-4 aspherical (27.2 - 56, f2.8-4)
Cannon EF 100-300mm 4.5-5.6 ultrasonic (160-480 f4.5-5.6)
Cannon crap lens that came with my EOS Rebel G, 35-80mm (56-128)
All of your filters would still work just fine, which, depending on what you've got, I know can be quite an investment.
All in all, you'd have (nearly) seamless coverage from 27.2mm out to 480mm with your existing glass.
I know it's not pocketable, but that's a nice bit of investment to be able to leverage and only be out the cost of the body on the DSLR, all to end up with a nice kit that's pretty flexible.
Pick up a pelican case on Steep and Cheap, toss it in the back of the 80, and you're ready to go