The first Hadeeth is recorded by Ibn 'Adi and Ad-Daraqutni with a chain to 'Abdullah bin 'Umar, may Allaah be pleased with them, and in these words: "Whoever performs Hajj but does not visit me has been rude to me." It is a weak narration, and some say that it is fabricated — a lie, because Muhammad bin An-Nu'man bin Shibl Al-Bahili related it from his father, and both of them are considered extremely weak. Ad-Daraqutni said, "The flaw in this chain is in the son of Nu'man and not in Nu'man himself." Al-Bazzar also recorded this Hadeeth, but in his chain is Ibrahim Al-Ghaffari, and he is weak. And finally, Al-Bayhaqi recorded this Hadeeth on the authority of 'Umar, and said, "His chain is unknown."
Ad-Daraqutni related the second Hadeeth as well, "From a man from the family of Hatib, from the Prophet (sallAllaahu `alayhi wa sallam)." With the wording mentioned above, and there is an unknown narrator in its chain. Abu Ya'la recorded it in his Musnad and Ibn 'Adi in his Kamil, but Hafs bin [Abu] Dawud is in their narration, and he is weak.
The third Hadeeth is attributed to Anas bin Malik, may Allaah be pleased with him, from the Prophet (sallAllaahu `alayhi wa sallam). However, Sulaiman bin Zaid Al-Ka'bi is one of its narrators, and he is weak. It is also reported with a chain to 'Umar, and in it is an unknown narrator.
With this, there are authentic Hadeeths that mention reflecting on death and the Hereafter, and supplicating for the dead. But every Hadeeth about visiting the Prophet's grave is weak, and some of them may even be fabricated. And whoever wants to visit graves in general or the grave of the Prophet (sallAllaahu `alayhi wa sallam) specifically following the guidelines of the Shari'ah — visiting to reflect on death and the Hereafter, and to supplicate for the dead; sending prayers to the Prophet (sallAllaahu `alayhi wa sallam), asking that Allaah Almighty be pleased with his two Companions; doing all of this without making a journey specifically to visit the grave — then his visit is legislated and it is hoped that he will be rewarded.
If one begins a journey specifically to visit any grave, even the Prophet's grave, then that is not allowed, for the Prophet (sallAllaahu `alayhi wa sallam) said,
لا تشد الرحال إلا إلى ثلاثة مساجد مسجدي هذا ومسجد الحرام ومسجد الأقصى
"Do not mount up except to three Masjids: Al-Masjid Al-Haram (in Makkah), this Masjid of mine (in Al-Madinah), and Al-Masjid Al-Aqsa (in Jerusalem)." (Al-Bukhari no. 1189 and Muslim no. 1397)
In yet another Hadeeth, the Prophet (sallAllaahu `alayhi wa sallam) said,
لا تتخذوا قبري عيدا ولا تجعلوا بيوتكم قبورا وحيثما كنتم فصلوا علي فإن صلاتكم تبلغني
"Do not make my grave an 'Id (a place that you habitually visit), and do not make your homes into graves (by not praying and remembering Allaah in them). And send Salah upon me, for verily your greetings reach me, no matter where you are." (Abu Dawud no. 2042 and Ahmad 2/367)
And Allaah Almighty knows best.
 The Shaykh, may Allaah have mercy upon him, is referring to narrations from the Prophet (sallAllaahu `alayhi wa sallam), not the narrations about the Companions, may Allaah be pleased with them, visiting the grave of the Prophet (sallAllaahu `alayhi wa sallam).
 To say, "Allaahumma salli 'ala Muhammadin wa 'ala ali Muhammad"