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Dictionary form: お礼{れい}をする

Transformed into ず-form: お礼をせず

Is お礼をせず used similar to the English expression "without thanking" which
itself can work as an adverb, e.g. I ate without thanking

わたしは **お礼をせず** 食{た}べた。
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grammar negation
Are verbs transformed into the ず-form used like adverbs?
Yes.

However, something to be careful of is 〜ず vs. 〜ずに:

> お役に立てず、申し訳ありません。  
>  "We apologize for being unable to serve you."
>
> ✗お役に立てずに、申し訳ありません。
>
> お礼も言わず（に）食べた。  
>  "He ate without saying his thanks."

I think you could say that 〜ず allows for exactly what なくて supports plus
adverbial modification, while 〜ずに only allows for adverbial modification.
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_Strictly speaking_ the answer is no. Just the -ず ending by itself won't
create anything _"adverb-like"_. That's why it takes the "ni" particle (-ずに),
to become adverb-like (just like 自由 _jiyuu_ is _freedom_ and 自由に _jiyuu ni_ is
_freely_ ).  
When you have a -ず ending you have a 連用形 ren'youkei, also known as ます形. It
follows a negative base (未然形 mizenkei), but the auxiliary verb isn't negative,
it's, as I said, a ren'youkei, meaning that it's more like a substantive than
an adverb.

When it takes "ni", it becomes "adverb-like", usually depicting the way
something is done. In this case it compares to a ないで ending.

Without "ni" it's like a なくて ending, meaning the sentence will be something
like "[subject] don't A, and [subject] do B".

It's pretty straightforward if you think about it. You can coordinate two
sentences with a て形 te-kei or with a 連用形 ren'youkei (in different contexts).
But what if you want to coordinate a negative sentence? The first one becomes
nakute, but the second one? You can use ず (、), because, yeah, it follows a
negative base (未然形 mizenkei), but it's a ren'youkei if it comes to the
auxiliary verb -ず.
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