Water bending and healing are the same art.
In answer to your question:
When they heal using water, are they bending? If not, can non-benders heal?
The division between Healing and Waterbending arises solely due to the sexist traditions of the Northen water tribe:
A female has not been seen in the military of the Northern Water Tribe due to their sexist traditions. However, the other Water Tribes did not follow these traditions as female waterbenders were seen defending the Southern Water Tribe during the Fire Nation raids and females of the Foggy Swamp Tribe were seen during the Invasion of the Fire Nation.
~Avatar wikia on the Water Tribes.
We see that Hama was a female water bender in the above link. There are also male healers in republic city.
Healing is a 'unique sub-set' of water bending. To remove it from the pool of talents a water bender can call on would be like saying blood and plantbending wasn't water bending.
The main difference is that water bending is external to a living being, and the subsets are internal. You manipulate the water inside a person whether it's via bloodbending or healing. The two are very similar in that respect.
So we have to answer,
Would [the healers from the northern water tribe] have been full-blown benders had it not been for the Northern Water Tribe traditions?
Yes, had they been given proper training I'm confident they would have been full waterbenders. They had simply been only allowed to focus on healing. I wouldn't be surprised if some of them had secretly water bended, even if by accident.
What powers does the water have?
None, in the same way the 'fire' that forms lightning in lightningbending, nor the motes of earth used in metalbending have any overt powers external to the bender.
Some water does have spiritual properties, which aids healing, but it's not essential. It helps because healing effects chi pathways, which, incidentally if blocked prevent all healing.