这主要是语法问题，但与语义和语体（register）也有一定关系。

1.在符合语法规范的要求下，并列结构的第二个（及此后的）分句需要用that引导，一般不可省略。如果不用连词that，句子结构产生歧义。例如：

    I know (that) he's cheating and I can't do anything about it.

Quirk认为这是一个有结构性歧义（Structural ambiguity）的句子，可能产生的解读有两种：

（1）I can't do anything about it与(that) he's cheating并列，均为I know的补足语（宾语）。为了让明确这一点，需要在第2个分句前添加连词that，写成：

I know (that) he's cheating and that I can't do anything about it.

(注：第二个分句只要有that，就可以表明与前一分句并列，即使and前有逗号。例如： Note that adjectives are placed after the indefinite pronoun rather than in front of it, and that you do not use a determiner-CCEG)

（2）I can't do anything about it与I know (that) he's cheating并列。但如果不对现有句子重新措辞，则无法确认这一解读。因此，通常须在两句之间添加标点符号，最好是分号。更明确的办法是把and省略(或替换成however)。例如：

I know that he's cheating; (however) I can't do anything about it. （CGOEL）

如此，则 I can't do anything about it完全脱离了I know的控制，成为与主句并列的独立句子。

因此，从语法规范性的角度看，把两个并列的宾语从句视为一体而将第二个that省略的做法通常不可行。这是主流英语语法共识。

2.在脱离上述并列结构的一般性用法中，非正式语体常省略连词that，正式语体通常保留。这是确定是否省略that的若干因素之一。受这一因素的影响，在并列结构中，第二个that省略的情况在非正式语体中有可能出现。但前提是：从语义就能判断并列结构处于从句层级而非主句层级。而非正式语体通常句式简短、语义明确，通常具有省略第二个that而不被误读的条件。例如：It is possible that you are right and I am wrong. 如此短小又意义自明的句子在and后再用一个that就极不自然了。

3.关于It soon dawned on me that many university professors are quite willing to share what they know, and they do take a genuine interest in their students’ intellectual development.

首先是这个句子的句源是否靠谱。看上去是一个本科生的毕业感想，英语用法不规范的可能性是存在的。严格地说，and they do take a genuine interest in their students’ intellectual development会被视为与主分句并列，而非与前面的that从句并列。这也就是说 many university professors are quite willing to share what they know（教授们愿意分享知识）是作者当时的认识，而they do take a genuine interest in their students’ intellectual development（他们关注学生的智力发展）则陈述了一个客观事实。

如果坚持they do take a genuine interest in their students’ intellectual development的语义已经明确与前面的that从句并列，所以that被省略。我认为无论从语法、语义还是语体角度看，都颇为牵强。

仅供参考。