Question 1: Yes
According to Sahih al Bukhari, one of the central collections of orally transmitted words and sayings by Muhammad, he did make this prophecy:
حَدَّثَنِي يَحْيَى بْنُ قَزَعَةَ، حَدَّثَنَا إِبْرَاهِيمُ بْنُ سَعْدٍ، عَنْ أَبِيهِ، عَنْ عُرْوَةَ، عَنْ عَائِشَةَ ـ رضى الله عنها ـ قَالَتْ دَعَا النَّبِيُّ صلى الله عليه وسلم فَاطِمَةَ ابْنَتَهُ فِي شَكْوَاهُ الَّذِي قُبِضَ فِيهِ، فَسَارَّهَا بِشَىْءٍ فَبَكَتْ، ثُمَّ دَعَاهَا، فَسَارَّهَا فَضَحِكَتْ، قَالَتْ فَسَأَلْتُهَا عَنْ ذَلِكَ. فَقَالَتْ سَارَّنِي النَّبِيُّ صلى الله عليه وسلم فَأَخْبَرَنِي أَنَّهُ يُقْبَضُ فِي وَجَعِهِ الَّذِي تُوُفِّيَ فِيهِ فَبَكَيْتُ، ثُمَّ سَارَّنِي فَأَخْبَرَنِي أَنِّي أَوَّلُ أَهْلِ بَيْتِهِ أَتْبَعُهُ فَضَحِكْتُ
The Prophet (ﷺ) in his fatal illness, called his daughter Fatima and told her a secret because of which she started weeping. Then he called her and told her another secret, and she started laughing. When I asked her about that, she replied, The Prophet (ﷺ) told me that he would die in his fatal illness, and so I wept, but then he secretly told me that from amongst his family, I would be the first to join him, and so I laughed."
Sahih al-Bukhari [61:130]
Question 2: Yes
It is correct that Fatima was the first one to die among Muhammad's household, the Ahl al-Bayt. There is a disagreement on her exact death date, but it is unanimously accepted by historians that she died in 632 CE (the same year in which Muhammad died on 8th of June). The next member of the Ahl al-Bayt to die was Muhammad's wife Maria al-Qibtiyya in 637 CE. Among Fatima's own family, the first one to die was her husband Ali ibn-e-Abi Talib in 661 CE.