What if I supply an Arduino mini with 5v 3a, is it 3a current will damage it? What is the best current value to supply?
The current rating of a power supply is only the amount of current it can supply not the current it will supply. As an Arduino by itself takes only about 40mA then anything will do.
I would use a 500mA or 1A capable supply as they are more forgiving on the electronics when you make mistakes. But if you don't make mistakes the power supply can be capable of any amount of current, it will not matter.
Current is pulled by the device, not pushed by the supply.
The light bulbs in your home don't get force fed 15-20 amps just because that's what the breaker can supply. They use what they need.