Can You Counter a Planeswalker Ability?

They're not spells, so you can't use regular counters.

Can loyalty abilities be countered?

Yes, it's correct. The rule you quoted implies that loyalty abilities are activated abilities. And since there is no rule that makes an exception for loyalty abilities, they can be countered like any other activated ability. Remember that countering an activated ability only stops it from resolving.

Can you disallow a planeswalker ability?

The Planeswalker's ultimate is an activated ability, so yes you would be able to counter that ability. You target the ability on the stack, though, not the Planeswaler. Disallow has no effect on the the Planeswalker.