Asteroids as portable gas stations?

Yes, the amount of ore in an asteroid is limited, and it is discovered when you approach it with any vessel.

Yes, the amount of ore in an asteroid is limited.

That said, IIRC, asteroids are typically something like 90% ore by mass, and a Convert-O-Tron 250 can turn all that ore into an equivalent mass of fuel.* So KSP asteroids are basically big fuel tanks floating in space, except that you need to bring some ISRU gear along to actually turn them into burnable fuel.

In fact, a Large Holding Tank filled with ore is pretty much equivalent to a smallish class B asteroid in terms of mass and ore content, and makes for a pretty convenient asteroid simulator if you want to test your asteroid mining craft without having to catch an actual asteroid. (The holding tank weighs 2 tons when empty, and holds up to 15 tons of ore, giving it an 88% ore mass fraction.) The only major difference is that you don't need a drill to get the ore from the tank.

So, yes, for a large craft, it's totally feasible to grab an asteroid along the way for extra fuel. For smaller vessels, the extra mass of the required ISRU gear (about 6 tons for a small drill, a small holding tank and a large converter) may not be worth it, compared to just launching with extra fuel. It's also entirely possible to use the fuel extracted from a passing asteroid to park it in low Kerbin orbit, and (especially with the help of some careful aerobraking) still have enough ore left to make it useful as an orbital refueling station.

For both purposes, it's more efficient to focus on the larger asteroids (class C and up), since a tiny class A asteroid might not even have enough ore to make up the fuel cost of sending a mining ship up to it. On the other hand, the biggest class E asteroids can be a real pain to steer, especially if using the low-thrust nuke engines (which you otherwise generally should use, since they're the most fuel-efficient ones). Physics warp can be useful for reducing apparent burn times, but it can also cause excessive wobbling.** Using the "autostrut" advanced tweakable is an effective, if perhaps slightly cheaty way to fix that, but a better long-term solution is to only target asteroids that are of a suitable size for your ship.

^{*) You should never use the smaller Convert-O-Tron 125 for asteroid mining, since it basically wastes 80% of the ore. Using the smaller drills is fine, though.}

^{**) I also suspect there's some kind of a physics bug with asteroids in KSP 1.2 that sometimes causes ships docked to large asteroids to wobble uncontrollably in weird and seemingly unphysical ways, or even to spontaneously explode. Autostrutting the asteroid to the "heaviest part" (i.e. presumably itself) seems to mostly fix it, with minimal effects on normal KSP physics. I haven't yet tested to see if the same thing still happens in KSP 1.3.}