I have always been told to only let birds bath with supervision, as there is some risk of them drowning. For that reason, I don't leave my tub in or attached to the cage. Room temp. water is best-- since i can't see your tub, I can't really comment on how deep.
Different birds prefer to bathe differently. Some like little tubs, others bathe in wet greens. Some birds love running water-- my little guy Joy loved to run under the kitchen sink while I held my arm out as a perch. I would offer the birds the tub in a neutral spot, maybe on top of the cage, and see how it goes.