This rule is informally called the “summoning sickness” rule. If you have a way of tapping, summoning sickness does not affect whether you can untap the creature to use it's ability. It only states that it cannot attack or tap to activate it's ability.

What phase does summoning sickness go away?

TL;DR: Summoning sickness wears off when the creature begins a turn under its current controller's control. If it changes controllers, it will be sick again.

Can you convoke with summoning sickness?

You can use creatures with summoning sickness to pay for Convoke costs. Convoke doesn't change a spell's mana cost or converted mana cost.