Du`a' after offering Salah

Q 1: During Jumu`ah (Friday) Prayer, a Khatib (preacher) mentioned that there is a Hadith which entails that a man stood up after the Imam (leader of congregational Prayer) pronounced Taslim (salutation of peace ending the Prayer), so `Umar said to him, "You, hypocrite! Sit down and pronounce Tasbih (saying: "Subhan Allah (Glory be to Allah)") before you go." The Messenger of Allah (peace be upon him) said to `Umar, "You have done well and followed our Sunnah." Is this Hadith Sahih (authentic), Hasan (a Hadith whose chain of narration contains a narrator with weak exactitude, but is free from eccentricity or blemish) or Mawdu` (fabricated)? I think it is Mawdu`, but I would like you to give me a Fatwa concerning the degree of this Hadith. May Allah reward you best! The same Khatib mentioned that there is a Hadith which entails that a believer should not want to come out of the Masjid (mosque), whereas a hypocrite is like a bird which is locked up in a cage, so it flies away as soon as the cage is opened. This metaphor describes the same situation, which is coming out of the Masjid immediately after the Imam finishes Salah without pronouncing Tasbih, Tahmid (saying: "Al-hamdu lillah (All praise is due to Allah)") and Takbir (saying: "Allahu Akbar (Allah is the Greatest)") thirty three times each, and Tahlil using the form "There is no God but Allah Who has no partner; to Him belong all the sovereignty and praise; and He is Omnipotent over everything", or other forms of Adhkar (invocations). Are Muslims considered sinners if they do not pronounce Tasbih after each Salah, but go to their jobs or (Part No. 7; Page No. 115) homes? Is there a difference between those who have an excuse which prevents them from Tasbih and those who don't? Is Tasbih a desirable Sunnah, or a Fard (obligation) like the obligatory Salah? Please advise and tell us the truth, may Allah guide you best!