No, it doesn't have to do with your example sentence being in 敬語 and the verb in 尊敬語. Note the link @broccoli facemask provides in their comment.
【接続】
名詞[辞書形]+のことで
【意味】
〜について
【例文】
①N1の文法のことで質問があります
②留学のことで相談したいんだけどいい?
③A:春休みに行きたいところはある?
B:そのことで話があるんだけど…
④お金のことで話さなきゃいけないことがあるの…
You don't see を in there for two reasons:
- 彼のことで = 彼のことについて
- を is omitted and if put back should mark 何か
彼のことで/について何か(を)ご存知ですか
何か is the content (内容) of the action "to know", and 彼のこと is the scope (範囲). Literally: "What do you know about him?" Here "what" is the direct object of "know", not "about him". See? English works the same way and also allows us to set a range and talk about the specific content with one verb plus some prepositions.
Let's take a quick look at your sentences at the end:
彼のことを知っていますか
This asks directly and specifically if the listener knows "him" (or something about "him" that the asker has in mind). Depending on the context it could either be "Do you know him?" "Do you know about him?" or even "Do you know what happened to him?"
彼のことで知っていますか
彼について知っていますか
These two sentences mean pretty much the same. Standalone and with nothing else, they sound a little strange to me.
何か彼について知っていますか
何か彼のことで知っていますか
sound more natural to me.