1)
We already know that she is Shlomo's mother!
1.
Kli Yakar #1: This shows how 'love' (Adoniyahu's desire for Avishag) reverses proper conduct. He should not have gone to Shlomo's mother, since he took the kingship from him!
2.
Kli Yakar #2: He went to her, for [if he can persuade her] he was sure that Shlomo will honor his mother and obey her words.
2)
Why did she say "ha'Shlomo Bo'echa"?
1.
Kli Yakar: She knew that Adoniyahu hated her due to the kingship (she helped persuade David to anoint Shlomo), and she feared lest he came to harm her.
