No, we don't. They're related in meaning, but we use them differently. Emergency is concrete - it's a countable noun and refers to a particular situation in a given moment. 'Come quickly! It's an emergency.' Urgency is more abstract - it's an uncountable noun, and is more of a sense or a feeling. 'I could hear there was an urgency in his voice.'

They are similar, but not usually the same. Urgency means something important that must be done quickly or soon. Emergency usually means a matter of life and death or a very serious situation.

No. You can have an urgent emergency. Urgent would be top priority.