Puppy, Kitten, and City.-preface

A lone dog trotted down a back alley of New York City. The dog’s fur was rough and mangled with all different kinds of dirt, mud,and bugs. The dog seemed to be searching for something along the dank and dirty sidewalk.
Suddenly, there was a rustling behind a pile of leaves. The dog’s eyes widened, and unlike most normal dogs, this little dog started to back up, obviously scared. He kept his eyes trained on the rustling pile of leaves, ready to bolt at last minute.
Out of the leaves, came a tiny kitten, matted and shriveled. The dog threw back his head, scoffing.
The dog continued on, but stopped when he heard a small and desperate noise.
“Mew,” the kitten cried. “Mew.” The kitten shot her huge vulnerable eyes at the dog. “Mew?”
The dog gave her sympathetic look, and then raced toward the kitten with new energy. He didn't need a tag-along.
“MEOOOOOW!”

Lol. Very interesting.

The dog’s fur was rough and mangled with all different kinds of dirt, mud, bugs.

Add an 'and' after mud.

Was this what I was afraid of? I deserve better challengers, he thought.

Put the thought in quotes or at least put in italics. I didn't know it was a thought until I reread it and saw the 'he thought'.

Okay, this was interesting. There isn't much to say since it was so short. The descriptions were well done...And....That's it, I guess.

I'm wondering what will happen next.

Nariel

Interesting start! I like the dog being the main character. Very intriguing.

A lone dog trotted down a back alley of New York City.

I'm not sure about mentioning New York City yet. Perhaps there would be a better way to introduce the dog without cutting straight to specific setting.

The dog seemed to be searching for something along the dank and dirty sidewalk.

Because of Show Don't Tell, perhaps it would be better to say something like, "The dog turned his head from side to side, sniffing under piles of rubbish", to suggest searching without explicitly saying it.

Suddenly, there was a rustling behind a pile of leaves. The dog’s eyes widened, and unlike most normal dogs, this little dog started to back up, obviously scared. He kept his eyes trained on the rustling pile of leaves, ready to bolt at last minute.

Good. "This little dog" is a bit awkward, perhaps "he" would be sufficient (as there's only one character at this stage, that wouldn't cause confusion).

Out of the leaves, came a tiny kitten, matted and shriveled.

I don't think you need a comma after "out of the leaves", but I could be wrong.

The dog started to continue searching for competition, but stopped when he heard a small and desperate noise.

"Started to continue" sounds a bit awkward, possibly "continued" would work alone.

The dog gave her a moment’s pity, and then forged on.

Again, I'd Show Don't Tell the pity, perhaps have him look in her direction, hang his head, hesitate for a moment before moving on.

He didn't need a tag-along.

Interesting ending; makes the reader wonder exactly what a dog would be doing that he "wouldn't need a tag-along".