yes, she said she wants to take ye yibo to somewhere and she drove him to 半山 (Bànshān)..(This place 半山 /Bànshān is translated to "foothills" or "Mountains" in subtitles)
The first day shen cheng arrived in pushui from England ,he went to his mother's place and a telephone call From police station came which said ''your mother's car is parked at Bànshān for 2 days,but your mother has been unreachable" It's obvious that ye haitang didn’t drive the car straight to the river to kill ye yibo and herself , because the car was shown unscathed and parked safely without any passengers near the riverside of Bànshān
This place is the same place where ye yibo went to dump ye haitang's mother's dead body in 1991 and the same very place where Tang xixun did the murder and went to prison.It's ironic how the name of this place stayed the same for 35 years when most of the names were changed including the cities and towns...What actually happened after she parked the car, we don’t know,maybe they both got into fights and got drowned ,as the time and manner of their deaths were the same,evening of November 8th, both dead bodies submerged into the river.... it could be ye yibo killing her or her killing him and committing suicide after that.or it could be accidental drowning for both.
About the bucket thing, there are 2 possibilities. November 9 is Tang yixun's birthday, she always visited his graveyard with a cake and a flower bucket on that day,so she could be buying the bucket for that purpose ,or maybe she bought it because of her son shen cheng's arrival like you said.