As has been mentioned in the comments, although you have gathered together quite a few definitions and examples of usage, a common thread runs through them all in terms of what 'figure out' means.
In a broad sense of the word, to figure out something means to understand something clearly, however, what differentiates it from similar expressions such as to realise something or to find out something is the process involved in coming to that conclusion.
When we realise something or find out something, we are able to comprehend something that we couldn't comprehend before, but that may not have been our intention. Realisations can often happen by chance - something may spark a connection, and you then see it in a new light.
For example:
I just realised that there's an arrow hidden between the E and the
x in the 'FedEx' logo!
Similarly with 'finding out something' - we may have come by the information passively.
I just found out that my girlfriend was cheating on me.
Figuring something out, on the other hand, implies a conscious effort to study a matter or situation and actively search for the answer or understanding of it. Rather than by chance or happenstance, it is normally done with a specific goal in mind.
For example:
- We need to figure out what went wrong.
- It took them ages to figure out how to assemble the chest of drawers.
- When you've figured the answer out, put your hand up.
- They spent 20 minutes figuring out how to open the box.
There is a broad range of examples above, but all of these exhibit intent to come to a conclusion.
Best Answer
This is a metaphor that combines a political reference and a sports reference.
In the sport of boxing, the opponents each have a corner where they go for rest between rounds of the fight. Referring to someone "in the ____ corner" is a reference to this "sport."
"Red" and "blue" in this context have to do with how in some past presidential election, TV news settled on coloring states that were voting for the Republican candidate in red, and the Democratic candidate in blue.
So what this whole phrase means is that Donald Trump is said to be expressing the Republican view.