Question 1: Yes!
Yes you are right. Holy Prophet (Peace be upon him) did make this prophecy. Its a pretty straight forward one and can be found in lot of books - even it is found in Bukhari itself, having an authentic chain of narrators:
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(RA) 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!
This is also correct as she(RA) was the first one to die among the Ehl-e-Bait as well: There is a disagreement on her exact death date, but its unanimously accepted by historians that she died in 632AD (the same year in which Prophet Muhammad (Peace be upon him) died on 8th of June)
And no other member of Ehl-e-Bait died as early as 637AD (Umm ul Momineen, Maria al-Qibtiyya(RA)). Among her own family too, the first one to meet her was her own husband, Ali ibn-e-Abi Talib(RA) and that too in 661AD).