Ahmad von Denffer addresses this briefly in his book "Ulum al-Qur'an". He says that in the mus-haf of Ubay b. Ka'b (ra), in addition to it having surahs in a different order, there were two additional chapters and one additional ayah. The first one is entitled "Al-Khal'" or "The Separation" and translates as:
O Allah, we seek your help and ask your forgiveness, and we praise you and we do not disbelieve in you. We separate from and leave who sins against you.
The second one, "Al-Hafd" or "The Haste" translates as
O Allah, we worship You and to You we pray and prostrate and to You we run and hasten to serve You. We hope for Your mercy and we fear Your punishment. Your punishment will certainly reach the disbelievers.
The forms for these two pieces are clearly akin to the numerous du'a qunut that have been narrated from the Prophet (saws) (in fact, parts of the above are verbatim in known adhkar).
There is another ayah found in Ubay (ra)'s mus-haf, which is the subject of discussion in this hadith:
Narrated Sahl bin Sa`d:
I heard Ibn Az-Zubair who was on the pulpit at Mecca, delivering a sermon, saying, "O men! The Prophet used to say, "If the son of Adam were given a valley full of gold, he would love to have a second one; and if he were given the second one, he would love to have a third, for nothing fills the belly of Adam's son except dust. And Allah forgives he who repents to Him." Ubai said, "We considered this as a saying from the Qur'an till the Sura (beginning with) 'The mutual rivalry for piling up of worldly things diverts you..' (102.1) was revealed."
So clearly Ubay (ra) was clear about what constituted Qur'an and what didn't, and if this non-Qur'anic statement was part of his mus-haf, then his mus-haf/notebook contained statements other than Qur'an and was meant for his own use, not as an official copy of the Qur'an.
This explains why this text occurred in his mus-haf - as for why these were not included in the Qur'an, it's because the surahs in the Qur'an were compiled according to the arrangement that was mutawatir or well-known to the Muslims at the time.
You can read the section of von Denffer's book where he discusses this here.
As for the rest of the claim that some of the Companions recited those du'as in prayer, qunut and other du'a are frequently recited in Fajr and Witr prayers. Additionally, there was no king Al-Malik Ibn Marwaan - there was however an Abdul Malik b. Marwan. This (mis)quoting calls into question the translation of this quote in Baihaqi. Even if the quote was translated accurately otherwise, it doesn't contradict anything since there is an established recitation of du'as in prayer.
In conclusion ; No, they were never part of the Qur'an. They were most likely supplications of the Prophet (saws) he used in the witr prayer.