Why is Jake even there?

If you overthink most stories they fall apart a little, but OK. Let's try this.
Say this is how it happened:
Alonzo had already chosen Jake as a candidate for his squad, based on his attributes, and his naiveté. He seemed like someone Alonzo could mold over time but was still a solid cop.

Then the incident in Vegas goes down with the Russian.

Hoyt's first day is already scheduled, probably as a genuine kind of interview type of trial. Alonzo takes a look at Hoyt, thinks about Roger's big stash and knows it the only way he can get that much money that quickly. So he takes a gamble and involves Hoyt, figuring the whole thing will go down better with the shooting board, the Three Wise Men, and the public if someone above suspicion does the shooting, a rookie with no blemishes on his record and no prior connections to himself or his team.

He concocts the idea of getting drugs in Hoyt's blood as a way to control him later after the raid on Roger's, since he's brining him along pretty fast. But, after the events of the day and Hoyt's reaction to the shooting of Roger, Alonzo feels that the threat of drugs in his blood won't be enough to keep the honest and righteous Hoyt in line or even quiet, and goes with the backup plan he'd put in place, just in case, and calls Smiley.

The way he looked at it, none of his guys catch any blame, his problem goes away, nobody will miss Roger, and Hoyt is expendable. If it worked out and Hoyt turned out to be as crooked as the rest of them, then he has a new team member who is shackled to him because they're involved in a murder together. If not, then Hoyt vanishes after the shooting, maybe gets found murdered in gangland and it's viewed as some kind of retribution for Roger.

That's the way the plan was supposed to work out. And the one thing that screwed Alonzo's plan up, namely his Plan B, was the one thing that Hoyt did during the day that Alonzo didn't control or put in place: Hoyt jumping out of the car and saving the girl who turned out to be Smiley's cousin. It was the one decent thing he did the whole day and it ended up saving his life.