1.  We split the boards down the middle to use them for the back of the shelves.

请问老师， 名词短语 the middle 作什么成分？其中 them 指的是 the boards吗？

down是介词，沿着。。。而下。the middle为名词，指the middle of the boards，即木板的中线。them是指代boards.

2.  We haven’t reserved a table, but we’ll take a chance on its not being full.

动名词复合结构 its not being full 整体 相当于一个名词作宾语吗？不容易理解，可否换做其他形式来表示，方便理解？

its not being full是动名词复合结构作介词on的宾语。相当于一个名词（或名词从句）。可以省略介词on, 改用that从句。

 We haven’t reserved a table, but we’ll take a chance that it（the restaurant) is not full.

that从句为chance的同位语从句。