Salam,
So here are points on the narrations you quoted, I will comment from the viewpoint that Muhammad (saw) is the LAST prophet for mankind until the descent of `Isa (as) at the end of times as is stated in the abovementioned reports.
["The sun has not risen over anyone better than Abu Bakr except if a Prophet comes."]
This correct translation is: "except a prophet" the word "comes" was added by the (Qadiyani) translator to hint towards a continuation of prophet-hood. So there is no evidence here for it.
[al-Mughira said: "It is sufficient to say 'Seal of the Prophets', for we were told that Jesus will come, so when he comes he (Jesus) is before him (Muhammad) and after him (Muhammad)."]
No doubt this might have caused some early Muslims confusion, I've explained above how there is no issue with saying Muhammad (saw) is the last prophet and that `Isa (as) only comes at the end of the world. I wouldn't factor anything happening in those days as being a norm. Additionally, the Prophet (saw) said "No messengers or prophets between us, `Isa will be my Khalifah on my nation."
Furthermore the narration is weak due to Mujalid.
["There is no Nubuwwah after me except the Mubashshiraat." It was said: "And what is the Mubashiraat, O Messenger of Allah?" He said: "The good Vision" or he said: "The pious Vision"]
No issue here as well, regular believers have visions from God and dreams. Prophets often recieved them in addition to certainty and infallibility and some other qualities and traits. This matter is a small blessing that remained from the traces of prophet-hood. The narration implies that prophet-hood as it was known has ended with Muhammad (saw).
[said to al-Abbas b. Abdil Muttalib: "Within you there is Nubuwwah and within you there is Mamlakah (Kingship)"]
That is true, banu Hashim were blessed by having the Prophet (saw) from among them and by recieving the kingdom and leadership during `Ali's reign and that of the `Abbasies as well as those who rule in our days.
[say: "There were in the nations before you Muhaddithun, so if there is anyone in my Ummah from among them then Umar b. al-Khattab is from among them." Ibn Wahb explained Muhaddithun as meaning Mulhamun]
True, there are people who have been blessed with success from God, he helps them utter words to benefit those around them and inspires them sometimes to take actions that bring humanity goodness. I've met such people on several occasions, pious friends of Allah.
In conclusion, I observe that the evidence of those believing in the sessation of prophet-hood after Muhammad (saw) till `Isa's (as) descent at the end of times is much stronger than the belief that regular prophet-hood is continous throughout the ages. This cancels out the prophet-hood of all those who claimed it for the past 1,400 years.