In this way, how to get started would be less formal and much more conversational than how to start. In your examples, 'this action got started' might be used especially in the us, but sounds. For example, let's start the engine and see if the car won't make that noise again. as for your two sentences, i agree that let's get started on building this table sounds a bit awkward, but.
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From that point things started to get complicated. Getting started implies a description of the process, get started is a suggestion to the reader to do so, obviously to be followed by instructions. The verb phrase get started:
I should start that sooner rather than that later.
However, the two are not always interchangeable. In the given example, yes, you can replace get started on with start. In let's get started, the starting point is in view and let's get going, you are on the starting point already. Here, the meaning of 'get' is 'become', or 'be' in the transformative rather than durative sense.
Don't get me started is a fixed phrase with a specific meaning. Aside from the formal/informal distinction, there is a slightly different meaning between. Moreover, there is a sense of extra involvement abundantly made. From that point things started to getting complicated.
To begin doing or working on something you (had) better get started if you want to.
From that point things started getting complicated. The whole thing is short for don't let me start to talk about x.
