reportedly removing masks from demonstrators they arrested：

是现在分词短语，作“伴随状语”，是谓语动词used 发生之后的次要动作。

reportedly 是副词，“据报道；据说”，修饰现在分词removing。

注：reportedly 不修饰used，used 的动作是实实在在地发生了，而removing 只是据说，没有亲眼所见。

Police used water cannon, tear gas and truncheons, reportedly removing masks from demonstrators they arrested, and a number of people were injured.

这里的V-ing分句（黑体部分）作何理解？

She took a note from her purse and slammed it down on the counter.

这里2个动作，显然不是同时发生的，而是略有小间隔，

但这样的情况，也可以用“分词分句/短语”来写：

Taking a note from her purse, she slammed it down on the counter.

类似地，

Opening the file, the detective took out a newspaper cutting.

注意一下动作的顺序：

She took a note from her purse, slamming it down on the counter. 

--- 这是很文学的语体。

请注意标红的字。

至于这是什么状语，你自己命个名字就行，别整这种没用的东西。