I link to a source for each of these; the text is a rewritten summary from the source.

Kathy Hochul had a career as a lawyer. She goes into politics in 2011.

1. Source: https://ballotpedia.org/Kathy_Hochul

Rep. Chris Lee (R) has a spot in the US House. He's hit with sexual allegations and resigns.

Kathy Hochul moves into his newly opened position via special election.

Next regular election she loses.

2. Source: https://ballotpedia.org/Robert_Duffy

Cuomo's Lieutenant Governor Robert Duffy is eligible for a second term but mysteriously decides not to run and leave politics completely. Cuomo chooses Kathy Hochu as his replacement and she moves into the spot.

Robert Duffy has no official allegations against him per google. But his not running for reelection "to spend more time with their family" is a generic excuse that sometimes turns out to be that they were blackmailed behind the scenes to stay out of the race. It's theoretically possible he was told to drop out of the race or he'd be accused of something sexual.

I Lieutenant Governor is similar to Vice President - you're not elected directly, if the person you're running with gets elected (Cuomo) you're in.

3. Source: https://www.nytimes.com/live/2021/08/10/nyregion/cuomo-resigns

Gov. Andrew M. Cuomo gets hit with sexual accusations and forced to resign. Kathy Hochul is assigned his position as Governor of New York (no election).