Would this "being paid while flying" scenario be considered "flying for compensation"?
The short answer is that you'd need a commercial certificate, and potentially a commercial operator's license, to do what you're describing. Standard disclaimer: IANAL, and if you wanted to do anything like this, you really should find an aviation lawyer and get answers from them in writing!

The rules that cover this are contained in FAR 61.113, quoted in relevant part:


  §61.113   Private pilot privileges and limitations: Pilot in command.
  
  (a) Except as provided in paragraphs (b) through (h) of this section, no person who holds a private pilot certificate may act as pilot in command of an aircraft that is carrying passengers or property for compensation or hire; nor may that person, for compensation or hire, act as pilot in command of an aircraft.
  
  (b) A private pilot may, for compensation or hire, act as pilot in command of an aircraft in connection with any business or employment if:
  
  
    (1) The flight is only incidental to that business or employment; and
    
    (2) The aircraft does not carry passengers or property for compensation or hire.
  
  
  (c) A private pilot may not pay less than the pro rata share of the operating expenses of a flight with passengers, provided the expenses involve only fuel, oil, airport expenditures, or rental fees. 


To point 1, the answer is no. You're being compensated for your time, which is spent flying. If you were alone, 61.113(b) would allow you to be compensated - but that only applies if the aircraft does not carry passengers. Since the 61.113(b) exclusion doesn't apply, you're stuck with 61.113(a)'s prohibition of flying for compensation or hire.

You make an interesting point at the beginning about salary covering strange hours. If you really were paid just for your productivity, and you had an arrangement where you could just go take a couple of hours off in the middle of the day to go flying (or any other personal activity) without burning leave or affecting your pay, you could claim that you weren't being compensated for your time. I might write to the local FSDO for clarity on that before actually doing it, though.

If you were an hourly worker (#3) and were not paid for the time you were flying, that would eliminate some of the compensation. Note, however, that the FAA constructs "compensation" very broadly, to include good-will from others. If you made this flight in order to make your boss happy, you're being compensated! The source of the incentive to fly would be important: if you and a co-worker had to get to a destination, and you suggested, "Let's fly there!", you're under much better grounds than if your boss told you to.

The second point would depend on your normal work situation. Let's look at two portions of the flight:


The portion from home to work, and from work back home: In this case, you could be compensated for the flight under 61.113(b) - both in salary and costs, as long as your employer has a policy of reimbursing staff for their commuting costs.
The "non-commuter" costs: You could ask for 50% of the costs of the flight under 61.113(c) (see Mangiamele interpretation) - but from your co-worker, not your employer. Where they get that money is up to them. One important point: 61.113(c) requires "bona fide common purpose" -- you and your co-worker must each have a reason, beyond the flight itself, for going to the destination airport. If you weren't required to meet the client (or do something else that requires you to physically be in Waco at that time), you could not even share the costs of flight. Note that you can't be paid or reimbursed for your 50% — you have to eat that cost yourself.


To #4, the owner of the plane doesn't matter that much here, for a private pilot. The question of who provides the plane becomes very important in common carriage, but that's a different question.

To your last point, I'm not a lawyer - but I imagine that being fired for refusing to take an illegal action would be wrongful termination.

Further Reading:

An FAA Article on cost sharing

Mangiamele Interpretation: a letter from the FAA Office of Legal Counsel discussing this topic