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1.我查阅了手头的近百部词典，结果跟你发现的一样，所有的there is no doubt that...和I have no doubt that...中的that都没有省略，而在单独的No doubt后却都没有that, 除了在《张道真英语用法词典》中的以下这个例句外：

No doubt that the region had a sombre beauty of its own.(Priestley)我认为，这一地区有一种属于自己的阴郁的美。

因此，你之前的认知是正确的，但在语法填空中发现有no doubt that，这是极其罕见的，在今后的考试中若能再次遇见，就填that。若不是要求填that，只是短文中有这次现象的出现，你理解句子和短文的意思即可，不必理会。

2.关于no doubt是不是there is no doubt的省略，答案是否定的；关于“名词词组副词用法”是正确的，证据如下：

《英语用法指南》(英语Michael Swan编著)第三版第377条 P558

no doubt表示probably(大概，很可能)，或者I suppose (我料想，我推测)，不表示certainly(肯定)。

No doubt it'll rain soon. 我看很快就要下雨了。

表示“肯定”可以用以下表达方式：there is no doubt that (正式)，without any doubt (正式)，certainly和definitely。

据此可知：

①表示probably与你所听到的“副词用法”相吻合；

②不表示肯定，也就是说，no doubt不会是表示“肯定”的there is no doubt that句型的省略)

我们再看看别的词典的解释：

(1)《朗文当代高级英语辞典》P444

No doubt and (more formally)doubtless can be used simply to mean "I think" or "I agree". No doubt及(更正式的)doubtless可仅用来表示“我认为，我想”或“我同意”：No doubt you'll be at the party tonight (=I expect you'll be there).我想你今晚去参加聚会吧。(键者注：此种解释与“指南“中的I suppose (我料想，我推测)相当)

(2)《牛津高阶英语词典》

no doubt
used when you are saying that sth is probable: No doubt she'll call us when she gets there.
used when you are saying that sth is certainly true: There's no doubt about /that...
(键者注：两种解释正好证明了两点：①可以证明no doubt是名词词组“副词用法”;②no doubt不是there is no doubt that 的省略，因为前者是表示“可能“，后者是表示”肯定“)

3.关于no wonder与It's no wonder

no wonder是I's no wonder的省略，两者后面的that可要可不要。请看以下证据：

(1)《牛津高阶英语词典》

(it's) no / little / small wonder (that) ... it is not surprising: 

It is little wonder (that) she was so upset. 

◆ (informal) No wonder you're tired, you've been walking for hours.

键者注：从(it's)可知，no wonder是it's no wonder的省略；

             从(informal)No wonder...可知，在非正式英语或口语(见下面朗文)中常省略It's。

(2)《朗文当代英语词典》

(it's) no/small/little wonder (that)
especially spoken used to say that you are not surprised by something
　No wonder you've got a headache, the amount you drank last night.

跟牛津的说明一致。

关于要不要that，再从词典中找来几例，供你参考：

It’s no wonder he never came back. (ELD) （省略了that）

It's no wonder that the children love to visit the farm. (DAI)(保留了that)

Brad was Jane's brother! No wonder he reminded me so much of her!（省略了that）

No wonder that he divorced her. (Chambers) (保留了that)

从所查这些词典中的大量例句给我的感觉来看，that略省的情况更多。