It depends. She keeps urging you to stay. They keep urging you to stay. Basically, if the noun doing the urging is plural, no "s", if the noun doing the urging is singular then it is keeps.

That is the correct spelling of "urging" (persuading).

The mother was urging the boy to do his homework now, and not procrastinate.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

Cleopatra married her first husband, her brother, in 47 BC at the urging of Julius Caesar.

No, it is not a conjunction. It is the present participle of the verb to urge and can be a verb or a noun (gerund). The participles urging and urged do not function well as adjectives.