Need help with comprehending these sentences from "A Man, Prologue."

Welcome! Glad you are here to learn. To start with the phrase "-Is this mean" should be "-Does this mean" but the rest of your questions are well written.

Whatever I wanted to get across he understood well, -Does this mean "Whatever I want to tell him he understood me well"? Yes; to get across means to communicate to another, to get information or a message across. This often signifies difficulty or importance.

in a seat somewhat removed from the bartender’s post. "somewhat removed from the bartender’s post" Somewhat does not mean very much or very little. It is a way of saying not too close or not too far. It is used here to mean not particularly or noticeably close to the bartender. In a story it generally means that you would not notice the distance until later when it might be important. Here I think it means the friend want the bartender not to hear what they are saying or wants to keep the talk private.

Despite his lean build he could definitely hold his liquor. Yes; holding one's liqueur means they can drink a lot with little ill-effect.

Kido-san had left vague in the interest of confidentiality, Partly; Leaving things vague is a means of shielding or avoiding the truth without actually lying. If things had to be specific then the truth of some of the actions would be more clear and perhaps embarrassing. Doing so for confidentiality means leaving out just enough facts to keep the individual's identity confidential.

--Below are sentences that I can't understand at all--

in a rich man to boot! Full sentence (Kaido-san told taxi drivers the route with warmth and care an admirable quality in this day and age and in a rich man to boot!) "...warmth and care, an admirable quality in this day and age and in a rich man to boot" I've added the comma which may help. The qualities of warmth and care are what he is talking about. He is calling it remarkable that such qualities can be found in a rich man. The term "to boot" is an expression meaning "Along with" or "with everything else". It connotes their surprise that these qualities are included.

involving as they did a number of moving incidents Full sentence (The other stories this lawyer related were surprising, involving as they did a number of moving incidents about which Kaido-san himself had never spoken.)
Moving incidents refers to emotionally significant moments that are difficult or embarrassing to relate as they tend to bring back the emotions.