Tire sizing can be pretty important.
The first thing to note is that tire sizing that you can run on a bicycle is a function of the rim (the width of it) and the frame (the tire needs a few mm to clear the frame so that it doesn't rub the frame). A rough rule of thumb is that you won't be able to more than 1 or 2 tire sizes up on a road bike from whats originally spec'd (that being said, there are a lot of road bikes which violate this).
The second thing to note is that an air filled tire absorbs shocks through its deformation. As Sheldon Brown notes, a properly inflated tire will absorb bumps, have low rolling resistance, not pinch flat (if you have a tube) and keep you in control. The proper inflation of a tire is a function of the size of the tire and the load (weight) it has to carry. For example, on a 26x2" tire, one person may use a pressure like 40 PSI whereas on a 700x20 tire, they may use a pressure of 100PSI+ for cruising around town. A larger tire will use a lower pressure than a smaller one to be properly inflated (and the lower pressure is usually more comfortable). A heavier load or one over rougher ground requires a larger tire. So, fat people shouldn't really be using 700x20 tires, for example (*).
Now onto the actual questions.
- Is it a terrible idea to try and use a road bike anywhere but pavement?
Depends on the road bike. Some are better at going on fire roads and other surfaces without significant drops than others. A larger tire size is helpful. Cyclocross bikes tend to be quite good at this sorta thing too.
- How much of a difference does a wider tire make while riding on pavement?
Depends on the tires. A larger slick tire can be better on pavement than a smaller knobby tire. Unless you're a racer, you probably won't see a penalty with buying a bike that ships with 25 mm tires vs 32 mm tires in performance; in fact, you'd probably get a bit more comfort out of it (bikes designed for bigger tires generally tend to be more comfortable as well).
Note that there are other factors which go into which bike to pick, and the best way is to get a test ride in conditions you'd ride in. That being said, walking paths near canals tend to be pretty smooth typically, so I wouldn't fret on a road bike.
(*) There are obviously other issues to tires for heavy loads; a tougher built bike is better in this case, particularly high spoke and well built wheels.