Does Joker kill this character in the end?

There is no way to know for sure.

We cannot even know for sure whether the meeting with the shrink happens before or after the scene with Joker running around with bloody foot prints, let alone whose blood it is.

The movie makes it unclear on purpose.

In short, other than the two listed daydreams, the rest of the story makes sense when true, and there is no additional narrative value derived from any of it having been a daydream.

It's very common to find answers on Joker that play the "it could be a daydream" card, but this is in my opinion a cop out. Yes, some things we saw did not in fact happen, but that does not lead to the conclusion that nothing happened.

There are two known untrue events:

• Arthur being invited onto the stage from the audience on Murray's show
• Arthur developing a relationship with Sophie

Both of these were explicitly shown to be daydreams.

• Arthur snaps back to reality, watching Murray's show with his mother
• Sophie barely knows Arthur, and we see her disappear from the earlier scenes Arthur imagined her in.

Everything else in this movie having happened makes for a coherent story, without any need to introduce more daydreams.

If you investigate if a major plot point was a daydream, it often leads to a web of connections whereby previous (or future) scenes which corroborate the information would have to be a daydream. This dramatically increases the amount of imagination needed, as these scenes tend to introduce new information that Arthur couldn't possibly have known, or involve new people that Arthur doesn't know or is interested in.

Some quick examples:

• Did he kill Murray live on TV? Well, if that was a dream, there are a lot of scenes he would've needed to imagine. It would also not quite explain the exploded riots (more so than before the shooting) in the street, which end up crashing his police car and temporarily freeing him.
• Did he kill the three guys? Well, then all scenes involving any mention of it, including the inspectors, and even going on Murray would all have to be imagined too, since Murray very much addresses this event. Therefore, the previous bullet point would also have to be imagined.
• Did he invent the gun? Well, then he couldn't have had killed the three guys, and then the above two bullet points would both have to have been imagined as well.
• Did he imagine Randall giving him the gun? Well, then where did he get it from, and why did Randall pre-emptively bring up the alleged gun buy attempt only after Arthur was caught carrying a gun? Furthermore, in the later scene where he kills Randall, Randall repeats the "you're my boy" statement from the gun-giving scene. If the first scene was imagined, the odds of Arthur imagining that quote are astronomical.

None of these plot points being day dreams makes sense from both a narrative point (it actually devalues the plot events) or a cinematographic point (none of this was even hinted at being false, as opposed to the two daydreams we know of).

The two daydreams he has revolve very specifically about people he knows, contains events that make him feel loved, and are very shy on additional details.

• He grew up watching Murray and his daydream makes it seem like he's looking up to him like a father figure. Notice also that when he confronts Thomas Wayne, who he believes to be his father, Arthur says he doesn't want anything other than "maybe a hug". Compared that to the Murray daydream, where Murray acknowledges Arthur's existence and hugs him. Imaginary Murray gives Arthur what he wants from his allegedly real father.
• Sophie was nice to him in real life (sharing the misery of a horrible building), and all subsequent daydreams have based Sophie's character on that one nice interaction between Arthur and Sophie. Notice how (imagined) Sophie has no character, other than being nice and loving to Arthur. No meaningful dialogue, no personality, nothing.

The major plot points, however, have way too much detail, are not in Arthur's favor, and are much too connected to other parts of the movie, for them to easily have been a day dream without detracting from the value of the narrative.

So, in short, other than the two listed daydreams, the rest of the story makes sense when true, and there is no additional narrative value derived from any of it having been a daydream.