The Magen Avraham (OC 494) suggests a reason for the minhag to stay up all night to learn on Shavuos. The midrash relates that the Bnei Yisrael slept in on the morning of kabbalas hatorah and had to be woken up. To rectify this, we stay up each Shavuos night, learning Torah, thus preparing ourselves to receive the Torah anew.
R’ Ephraim Greenblatt (Rivevos Ephraim 8:500:12) writes that if staying up will prevent them from being able to daven properly, they should rather go to sleep and daven later.
Staying up all night, however, poses a couple of halachic issues with regards to saying certain berachos.
The Mishna Berura (47:28) writes that there is a machlokes as to whether one who hasn’t slept may recite the Birchos Hatorah. He writes that one should either listen to someone else (who has slept) recite the berachos or have intent to fulfil the beracha before learning while saying ahava rabba. One doing this should learn immediately after Shacharis. The Mishna Berura writes that R’ Akiva Eiger held that one who had slept during the day for at least half an hour is considered to have slept and may say all the berachos as usual.
Many sefardim follow the Kaf Hachaim (OC 46:49) and R’ Ovadia Yosef (Yabia Omer 5:OC 6; Yechaveh Daas 3:33) and recite the berachos regardless (See Rivevos Ephraim 8:500:15).
The Mishna Berura (4:30) writes that while there is a machlokes as to whether one should say al netilas yadayim after washing one’s hands, if one excuses themselves, they can say the beracha. R’ Ephraim Greenblatt (Rivevos Ephraim 8:500:13) writes that sefardim should rather listen to someone else recite the beracha.
One shouldn’t say the beracha on tzitzis, though should either hear the beracha from someone else (who says it on either their tzitzis or tallis) or borrow a tallis and recite the beracha on that (Mishna Berura 8:42; Rivevos Ephraim 8:500:18).
As the berachos of Elokai Neshama and hama'avir sheina both refer to waking up, one should ideally listen to someone else who had slept, say them. Failing that, the Mishna Berura (47:30) writes that one should say these berachos without Hashem’s name. R’ Ephraim Greenblatt (Rivevos Ephraim 8:500:14) writes that sefardim may say these berachos regardless.
As for the other birchos hashachar, the Mishna Berura (46:22) writes that one should say them all oneself, as they are all berachos praising Hashem and not dependent on personal circumstances (See Kaf Hachaim OC 46:50; Rivevos Ephraim 8:500:16).