ずに is the formal allternative to ~ないで (usable in those cases where ~ないでmeans 'and'). I hadn't realised you could use the ず form without に before this :-)

is it possible to name all the English equivalents of に as a particle? So far I got "to", "for" and "by".

I don't personally see much point in trying to make up a list like that -- you'll only get bitten by the edge cases where the two languages don't match properly. The Dictionary of Basic Japanese Grammar divides the various uses of に as follows, starting from the core idea of 'contact':

4. Direct Contact -- 黒板に絵を描いた。
6. Locational existence (derived from 4) -- ここに電話がある。
7. Direction -- 私はロンドンに行った。
5. Purpose (derived from 7) -- 僕は魚を買いに行った。
2. Indirect object -- 太郎は花子に本を貸した。
3. Source/Agent -- ボブはメアリーに切符をもらった。
1. Point of time 二時に友達が来た。

(The numbering and ordering is as per the DBJG; I don't know why they mix it up like that.)