I have a couple of problems with my brakes on my Specialized Hardrock with cantilever brakes.

1) One of the brake arms doesn't move when I squeeze the lever. Thus, only one brake pad engages the rim. The arm is a bit "stiff", but I can move it back and forth using my hand.

2) When I turn left, the brake pad is hitting the rim. When I go straight, there is no friction.

Thoughts?

1. When you say "doesn't move", do you mean initially when you start squeezing the lever, and one pad moves before the other, or the second pad never moves, never touches the rim? If one of the arms has more friction, it will lag, but if it's not moving at all, it has more than "more friction".
2. Ditto on too short housing - I assume it's the rear brake you have an issue with?

1. The arm never moves. The other arm moves much more freely.

2. It's actually the front brake. I can hear the pad rubbing on the rim when I turn left.

Shimano canty's?

Check the round gray plastic covers where the brakes attach to the frame. Do you see or feel a crack?
That's common with those older Shimano canty's. The gray plastic cover contains the return spring. When it cracks, the return spring loses it's tension and that brake arm doesn't function properly anymore.

The "A" solution is to replace the brakes. I generally replace them with lower end Shimano V-brakes. You have to replace the brake levers too.

The "B" solution is to replace the spring covers. Shimano used to warranty them at no cost to bike shops but I don't know if that's still true today or not.

The "C" solution it to wrap a zip tie or some mechanic's wire around the cracked spring cover.

Problem 1

First of all, check that the spring on both sides is well adjusted. See "Spring Adjustment" on this page by Sheldon Brown.

If not, it might be slightly more serious, but not terminal. Indeed, that's a problem I have at the end of each winter. Fortunately, the fix is cheap.
– Remove the cable from the arms so it's not under tension.
– Unscrew the pivot bolt (either allen key or wrench).
– Remove the offending arm.
– Clean thoroughly the pivot and the place in the arm where the pivot sits. It's probably full of grit, the grease is dry... I typically use WD-40 to clean both parts and let dry completely.
– Check that the spring isn't broken (unlikely, but it happens, in which case you'll need a new spring).
– Grease the pivot liberally, re-install the arm.
– Do the same with the other arm, because the problem will arise within 2 weeks.


Problem 2 – front wheel problem

At first reflection, it sounds strange, because cable length doesn't change when you turn the wheel. I see a few possible causes:
– Springs way too weak (see "Spring adjustment" above).
– The rear brake or one of the shifter cables interfere with the front cable and moves it sideways; in doing so, it "de-centres" the brake.
– It happens only when you turn the wheel completely, in which case it's simply because the brake arm touches the downtube. Not a real problem in real-life.
– Your front wheel wobbles, has very loose cones, slack spokes, etc. so it's actually the rim that moves sideways when you turn. It's unlikely, but I've seen it.