Can I use Rings of Brighthearth for my Plansewalker's abilities?
I know it says "A mana ability is an ability that (1) isn't a loyalty ability" so I'm guessing it CAN be used to put loyalty counters on Planeswalkers?

Rings of Brighthearth can indeed be used to copy your planeswalker activations, but doing so won't put any more loyalty counters on those planeswalkers.

When you copy an ability, you place a copy of that ability directly onto the stack, without paying any of the costs that would normally be associated with activating it, in the same way that if you Twincast your opponent's spell, you don't also have to pay the cost of casting that spell.

Putting the counters onto a planeswalker to use their positive loyalty abilities (, , and so on) is one of the costs of activating the ability in the first place--it's not part of the ability's effect. So copying the ability bypasses that cost--you get the effect a second time, but you don't also get twice the counters.

Can I copy the effect more than once in a single turn?

Not from a single Rings of Brighthearth and/or a single activation. Rings has a triggered ability that happens only once per activation. If you have multiple Rings out, though, then you can copy the activated ability once per Rings and assuming you have the mana to pay. So, if you had two Rings out and the capability to pay 4 mana total, you could copy the activated ability twice.