Hashirama may have transferred some of his senjutsu chakra for his binding jutsu to work, but I don't think it will lead to Sasuke getting Sage Mode. He already has Sage chakra in some form via Hagoromo.
Nope, You can't give SM to someone, you can only gain it through training. That's manga fact as Kabuto stated when Sasuke asked him if he was attacked by Orochimaru's Curse mark and Kabuto said the Curse mark is a pretension trick and stated that SM power can only be obtained through training. If Hashirama could give his SM to someone from his family and pass it down his family line so Tsunade would have Hashirama's SM if that was possible.
Also They were only thrown into the ground, not from a high grown, but just from where they were held by Kaguya's hair. Kakashi and Sakura could have took the hit Naruto and Sasuke got from being thrown like that, it wasn't anything special. Juubito in fact did more damage when he threw Naruto and Sasuke from a higher attitude hard enough to had them get out of them forms (Complete Susanoo and full BSM). They still can't say both can take the same damage since you also have to factor in that Naruto has been fighting constantly for a day or two plus was in intense training prior to entering the war so it's not like he would be at full power as Sasuke would have. Also Naruto wasn't at full power since he had to borrow some chakra from Yin Kurama, gave out a lot of chakra to the alliance which include Sasuke, then went on to fight Obito where his chakra was absorbed by the god tree and he had to borrow more chakra from Yin Kurama so he was never at full power in his fight with Juubito.
Sasuke can't obtain SM, if you mean gaining it like Madara, he would need the knowledge to do it as well which Madara did have while Sasuke did not have. He didn't even know that much about Natural energy and had to remember what Kabuto said after Naruto explained to him. Hashirama couldn't have gave Sasuke SM since he stated himself that Madara took his Senjutsu and that he could not knead more chakra since the rods stopped him so he gave Sasuke some of his regular chakra. To gain SM you need to balance your chakra with natural energy. The SM that Madara took wasn't on his face, it waso n the Hashirama's face on his chest, Sasuke has not SM markings either so your post lacks proof and goes against manga facts.
The SM that Madara took wasn't on his face, it waso n the Hashirama's face on his chest, Sasuke has not SM markings either so your post lacks proof and goes against manga facts.
hashi gave him his remaining chakra and jutsu that can bind madara in repsond to madara taking his senjutsu. so probably sasuke has hashi's senjutsu now