1. Believe it or not, his discovery has created a stir in the scientific circle.
——这不是虚拟语气。Believe it or not, 是评注性状语。说的通俗一点，就是“评论性状语”或“注解性状语”。它是独立于句子之外的一个状语成分。就是说，它不是修饰句子中的某个词语，而是修饰整个句子，它是说话者对所言之事所作的评论、评价或注解。“评注性状语”，一般放在句首。主要由副词（短语）、不定式短语、现在分词短语充当，还有一些其他情况。
Believe it or not, his discovery has created a stir in the scientific circle.
信不信由你，他的发现在科学界引起了轰动。
Luckily, he didn't die in the accident.
luckily 是评注性状语，是说话人对下句的评价。
To make things worse，many of the men have gone off to cities in search of higher pay，leaving women from nearby villages to carry on with the work.
To make things worse 是评注性状语，是说话人对下句的评论。
Personally speaking, it's a good idea. 就我个人而言，我认为这是个好观点。
Personally speaking 是评注性状语，是说话人对下句的看法。
 
2. Be rich or poor, I will marry him anyway.
——这是Whether he is rich or poor 的简略说法，表示让步，是虚拟语气形式。不过，这个句子有误，漏掉了 he，应为：Be he rich or poor, I will marry him anyway.
这是古英语遗留下来的一种倒装式虚拟语气，Be 动词用原形。常用于if 条件句和 whether，(even)  though引导的让步从句，网友的句子，是省略了连词的倒装句，后面有论述。现代英语，只有非常正式的文体才用。不过，在美国英语中，这种虚拟用法经常见到。又如：

If it be right, do it boldly. 假如觉得是对的，就大胆地去做
If this be the case, he deserves you. 当真这样的话，他的确配得上你。
Leave out my name from the gift if it be a burden, but keep my song. 假如我的名字成了负累，就将它从这礼物上拭去吧，但请保留我的诗歌。
If this be the price of the child's sympathy, I cannot pay it. 如果这是换取孩子同情的代价，我是不能支付的。
Whether he be rich or poor, I will marry him all the same.
无论他是富还是穷，我都要嫁给他。

Home is home, even though it be ever so homely. 家就是家，即使它是多么平凡。

◆如果省略连词，句子要倒装：
Be he rich or poor, I will marry him.
不论他是否有钱，我都要嫁给他。
Be she poor or rich, I will take her as my wife.
不论是贫穷还是富有，我都会娶她为妻。
She exhibits the idealism common to many advocates of sustainability, be it in food or in energy.
她展示了许多可持续发展倡导者所共有的理想主义，无论是在食品还是能源领域。
She enjoys watching dance, be it ballet, national minority dancing or any other sort of dancing.
(= whether it is ballet...)
芭蕾舞也好，民族舞也好，随便什么舞她都喜欢看。
来自《现代汉英综合大词典》
Speak frankly, I pray you, be it for life or death. " 
(= whether it is for life or death)
我请你坦率地讲出来, 不管我是该活还是该死。”
来自英汉文学 - 红字
Be it a painting, a photograph, a poem or a novel, if you created it, you own it and it’s the copyright law itself that assures that ownership.
(= whether it is a painting...)
不管是一张画，一张相片，一首诗或是一本小说，如果你创造了它，你就拥有了它，而它的版权法本身就保证其所有权。
Home is home, be it ever so homely.
(= even though it is ever so homely)
家就是家，即使它是多么平凡。

归纳起来，网友这个句子可以有3种表达：
Whether he is rich or poor, I will marry him.
Whether he be rich or poor, I will marry him.
Be he rich or poor, I will marry him.