Will Rings of Brighthearth + The Chain Veil let me use Chain Veil's ability more than once?

Question

I have a question in regards of using The Chain Veil's Activated Ability:

{4}, {T}: For each planeswalker you control, you may activate one of its loyalty abilities once this turn as though none of its loyalty abilities have been activated this turn.

Can I use Rings of Brighthearth copying ability

Whenever you activate an ability, if it isn't a mana ability, you may pay {2}. If you do, copy that ability. You may choose new targets for the copy.

to use Chain's ability more than once in one turn?

Answer 1

Yes. The sequence would/could be as follows:

1. You activate The Chain Veil, and put its ability on the stack.
2. The ring's triggered ability triggers, and is put on the stack.
3. (after all players pass priority) The triggered ability resolves, you are given the choice to pay 2 and copy the ability, and the copy is placed on the stack.
4. (players pass priority) The copied effect resolves.
5. (players pass priority) The original effect resolves.

A ruling from Gatherer confirms that The Chain Veil's ability can be repeated, i.e. each effect allows an additional loyalty ability to be played, on top of the single normal use.

Each additional time The Chain Veil’s last ability resolves will allow you to activate a loyalty ability of each planeswalker you control an additional time. For example, if you activate The Chain Veil’s last ability, untap it, then activate it again, you can activate a loyalty ability of a planeswalker you control three times that turn.

Answer 2

Yes, you can use Rings of Brighthearth's ability to copy The Chain Veil's ability, and yes, that will allow you to activate another additional loyalty ability of each Planeswalker you control. A ruling on The Chain Veil says

Each additional time The Chain Veil’s last ability resolves will allow you to activate a loyalty ability of each planeswalker you control an additional time. For example, if you activate The Chain Veil’s last ability, untap it, then activate it again, you can activate a loyalty ability of a planeswalker you control three times that turn.

Answer 3

Not only can you copy the chain veil but for each copy of the chain veil your planeswalker increases or decreases loyalty counters according to the ability you choose.

Example, if you picked a +1 ability and the chain veil activated twice, your Planeswalker will get 2 additional loyalty counters.