In spring 2013, Barron added MasteringChemistry prelecture homework assignments: short, untimed assignments due before lecture designed to help increase student awareness about new topics and to identify student misconceptions and areas of misunderstanding. Barron tells students it is okay to be confused after doing the homework because the concepts will become clear during class.
To Get is the normal form and would be best for everyday use. However book titles are allowed artistic license. Regarding the to Getting forms, I would think of it this way: 1) The Essential Guide to (doing something) + (which is) Getting Your Book Published and 2) The Guide to (doing something) + (which is) Getting it On In these examples each phrase can stand on its own, but you have follow ...
Yes there is a difference. Trees are getting cut down refers to an action that is in progress. Someone is cutting the trees. It is the form encountered more frequently. Trees getting cut down can be used in the context of an intended action. To explain: If someone has decided to cut some trees down, they may be referred to as the trees getting cut down. The trees are still standing at the time ...
This job’s really getting to me. Idiom "get to grips with" I'm trying to get to grips with the new software. As for your last example: He’s refusing to testify. I think the Mob got to him. You could perhaps explain this as a combination of Arriving somewhere and annoying. Or you could just say it's another phrasal verb with a special meaning.