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It was an August night in the Nevada desert

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This is a great scene setter. I get an idea of the narrator's mood without him even saying another word just by describing this hot, dry location.

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Seeing the Grim Reaper playing the slots disconcerted me, but I needed a phone, and badly. My car had broken down, and I hadn't had cell reception since driving through Hawthorne. Under any other circumstances I probably would have left immediately, but at this point I took Death’s presence as one more display in the parade of oddity my day had become.

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A great way to lead into the rest of the story. He see's the Grim Reaper but he has more important things to worry about, at the moment.

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Death was the first thing I saw when I entered the bar.

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This would need a little reworking
I saw Death when I entered the bar

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He was at the slot machines by the restroom door, plugging quarters into the slot and muttering to himself through his fleshless jaws. He was clothed in a tattered, black cowl, dusty with age and decay; his sickle leaned blade-upwards against the wall beside him.

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I like this here. This description of the Reaper a makes me wonder about him and I am eager on what happens next.
A little rearranging I think you have a great hook going.

A couple of comments on your thirteen:

1) Why are you using so many adjectives to describe what the Grim Reaper looks like? Doesn't everyone already assume that he's a skeleton in a black robe unless you tell them otherwise? I think this passage would be more effective if you either described what the robe smells like (grave dirt, rotting meat, flowers) or if you give him a new outfit (biker's jacket, Armani suit, black tank top and jeans).

muttering to himself How do you know Death was muttering to himself? Perhaps just saying that the was muttering or mumbling? Let us wonder about his muttering.

The description of Death sounds a bit contrived. You stop the story's progress to insert a physical description. How about something like "I knew it was Death by the tattered, black cowl that was dusty with age and decay. However, the sythe was the real give away." Incidentally, Death is generally identified with a sythe which is a long handled blade for cutting grain. A sickle is a short handled cutting implement with a curved blade.