First things first: you can make any change to your ticket even after travel has commenced, but it will cost significantly more than if you'd bought tickets for the new itinerary beforehand. In general, it is a very costly thing to do.
The method of calculation is simple: all segments that are changed from the original itinerary are considered, the price difference between the full fare on new segments and the old segments are charged. If there are completely new segments on the new itinerary, then you are charged the standard fare for these segments in full. No refunds, however, will be made, unless it is for an unused section over 101 kilometers.
Say you are going from Tokyo to Akita. You change your mind halfway, and decide to go to Shin-Aomori instead. If you change your ticket beforehand, you pay for Tokyo to Shin-Aomori, but if you change halfway, you'd have to pay (Tokyo~Akita) + max(Morioka~Shin-Aomori - Morioka~Akita, 0) = 10,780 yen (as opposed to 10,340 for a ticket from Tokyo to Shin-Aomori).
For certain short trips, you're simply charged the price difference between the old itinerary and the new itinerary (meaning that it is effectively the same as refunding the ticket and buying a new one).
Your case is convoluted, but there is nothing wrong with the change whatsoever. I would just ask the Midori-no-madoguchi at your destination (if there exists one), because they have access to the unified JR MARS ticketing system. Not all other counters, much less conductors on trains, have access to that system, but a staff with access to the MARS system can make any change of travel segment. If no station on your intended route have a midori-no-madoguchi (very rare case), I guess you may be out of luck.
Are changes made after the start of travel essentially the same as fare adjustment at the arrival station, just that fare adjustment at the arrival station is an ex post facto change?
Yes. In fact, fare adjustment (norikoshi) is just the most common form of change after commencement of travel.
How many times are you allowed to make changes after the start of travel?
I don't think there is a limit, and no limit is given in the Conditions of Passenger Carriage (Article 249 deals with such changes). But considering that essentially you are just buying a new ticket price-wise, I don't see why they will set a limit; if you make more changes, they simply charge you more in effect.
Where will the changes made after the start of travel be handled?
There is no specification, but I think a Midori-no-madoguchi would be the best bet.
At stations with a Reserved Seat Ticket Vending Machine inside the ticket gate, when can and cannot such changes be made at those vending machines?
I don't think you can make any changes on automated machines. You'd have to see a JR employee.
Are the staff allowed to refuse changes?
Yes.
If, on the other hand, I were using special services like the smart-ex service or the Shinkansen e-ticket service, would I be able to make changes in a similar way
No, special tickets do not generally allow changes. But in case of doubt, ask a staff member.
I wonder if, had I presented my post-change basic fare ticket (to Tokyo Ward Area) at the fare adjustment office at Tokyo station, would I have been able to change the fare ticket for a second time such that the arrival station is Musashi-Sakai?
You should be. But you will still be charged 170 yen (the fare from Nishi-Ogikubo to Musashi-Sakai), as changing the ticket at the fare adjustment office is the same as norikoshi.
In this case, with this second-time changed basic fare ticket, would I have been able to make stopovers at intermediate stations like Yotsuya, Shinjuku and Kichijoji?
You should be, as the destination is not within the Tokyo Special Ward Area.
