All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
In regard to your first question, Ibn katheer - trying to explain in his “Tafseer” why Allaah linked between the Jews describing their Lord as being poor and His mentioning that they killed the Prophets after that – said that Allaah clarified their (the Jews') misconduct with Allaah and their misconduct with His Prophets. Ibn Katheer when interpreting the saying of Allaah (which means): {We will record what they said.}, said: "This is a threat and a promise of punishment, it is for this reason that Allaah linked it with his statement {…and their killing of the Prophets without right…} which means that this is what they say about Allaah and this is their treatment of the Prophets of Allaah and Allaah will punish them for this a severe punishment.”
As regards your second question, the answer is that the scholars stated many reasons for mentioning the word ‘right’ in the saying of Allaah (which means) {…and their killing of the Prophets without right…} among which are the following:
1- Mentioning the word ‘right’ indicates that they killed them while killing them was not right in their belief and imagination; rather, they knew that it was wrong.
2- The word ‘right’ is mentioned here to emphasize the meaning like the saying of Allaah (which means): {And whoever invokes besides Allaah another deity for which he has no proof—}[Quran 23:117] It is known that whoever claims that there is another god besides Allaah he has no proof and he will not have any evidence about this in the future, so Allaah mentioned the word ‘proof’ to emphasize the meaning and it is the same thing here with the word ‘right’.
3- If Allaah were to blame them just for the killing, they would say that Allaah also kills, so Allaah, the Exalted and the Most High, clarified that He kills with right but others (like the Jews) kill without any right to do so.
As regards the third question, the answer for it is that the statement of the Prophet is confirmed in Al-Bukhari in the same Hadeeth: "You never know, it might be that Allaah looked at those who attended the battle of Badr and said, "O people of Badr, do what you like, as you deserve Paradise.”
Ibn Hajar said in “Fat’h Al-Baari”: “What is meant here is to give evidence for the excellence of the people of Badr by the above saying of the Prophet and this is a great glad tidings that is not given to other than them and the information was transmitted about this with some other wordings like “for I have forgiven you'' and “as you deserve Paradise”.
"Al-Aqeedah At-Taahawiyyah", after mentioning the excellence of the people of Badr, reads: “Despite the existence of many such texts, Ahlus-Sunnah do not testify that a particular person is from the people of Paradise except those whom the Prophet testified that they are from the people of Paradise, like the ten persons who were given glad tidings to enter Paradise, Al-Hassan and Al-Hussein as well as Thaabit Ibn Qays all."
Allaah Knows best.