朱雀 is already written like this in Chinese, so it is unlikely that the kanji writing would vary. In my (limited) experience this only happens when the word is of Japanese origin and a kanji had to be found for it in Japan.

As for the meaning of 子 when read -su: from the examples I can think of it is only used as a meaningless nominal suffix imported from Chinese, like in 椅子，様子 ...

Probably, the pronunciation of "su" isn't used to mean "child" but there might be some in very old Japanese which I don't know.

Usually, "shi" is used to mean "child"

男子（だんし danshi) -男の子　boy

女子（じょし jyoshi）－女の子　girl

養子（ようし youshi）－adopted child 

As you know、 朱雀　isn't a sparrow. It's a phoenix in red color. 

朱雀　also pronunced "syuzaku" or  "syujyaku"

Personally, I read it "syujyaku" because "syu" expresses the color of the phoenix.　

I 'll give you another example for "子” for "Su"

様子(ようす）:  It's called "you su"  means situation.

彼はおどろいた様子でした。- he seemed to be surprised.

どんな様子か見てみよう。- Let's see how's it going.

Suzaku is written as "朱雀".

As you said ”朱(su)" means the color vermilion, and "雀(zaku)" means sparrow.

You can not use "子(child)" in this case.

子 is usually pronounced as "Shi", and it's very rare to read as "su". (ex. 扇子(Sen-su))