Dear A Class:

You wrote:

>Which of these is it correct?
>Some materials could be mined from the moon and sent back to Earth for processing , or , to be processed , or, in order to process…

>In other words, I want to know is there any difference if we use ‘for + ing’ or ‘infinitive’ or ‘in order to’ to say the reason of an action?

The first two are OK. You would have to put the third one in the passive voice to make sense: “in order to be processed.” After all, the mined materials are not doing the processing!

>Another example: She did her homework quickly for watching TV / in order to watch TV.

The first one does not make sense. Homework does not assist us in watching TV. Perhaps “She wore glasses for watching TV” (i.e., she wore glasses to assist in watching TV).
The second one is fine.