1. Not a trick question. 

2.  Just trying to anticipate the next battle in the religious and political war.  

1.  I was going to take a stab at this even if this was a trick question, @IRonMaN 's warning notwithstanding.   If @IRonMaN was correct, in my sheer imagination or recollection, @BobKamman 's M.O. was to hook and reel after TWO bites.  @qbteachmt 's was the first.  Here cometh the second.

2.  Honestly I have NO idea what "the religious and political war" is all about.  I'm the kind who would begin to think about "investing" in Bitcoin after it hit $45K...

I agree with@qbteachmt that "That Pub 502 quote and this question as posed are confusing employer medical reimbursement provisions under an employer health reimbursement provision and the tax regulation allowance for medical travel, which is an individual's provision and doesn't require there be an employer involved at all."

That part of Pub 502 clarifies "Transportation" further.  Reg 1.213-1(e) (1)(iv) is very brief.  That part of Pub 502, for example, caps the out of town stay at $50 per night, and echoes the "spirit" of the Regs.

An MERP is not a section 125 plan, cafeteria plan, or flexible spending account.  Rather, it is a section 105 plan, like a health reimbursement arrangement (HRA).

Regs. 1.105-11(b)(1)(i):

Those are sometimes referred to as "Section 105 Plan".  Notice how the Regs REQUIRE a separate written plan.  

Thus, you may have two companies, one with and the other without, a separate written plan, to have different treatments on medical reimbursements.  Actually, upon audit, the treatment (for both the company and the employee i.e. wages levels) on the reimbursements made by a company WITHOUT a written plan  will be at the mercy of the auditor.  I imagine the deductibility by the company would likely be subject to the "ordinary and necessary" tests, and the reimbursements would be taxable to the employee as wages.  

Going back to Bob's original post as drafted, my take is that the answer would hinge on whether the company has a SEPARATE WRITTEN PLAN.

If NO:  Possible deduction by company.  Taxable to employee as wages - all of it - as taxable fringe benefits.

If YES: You'd have to look at the plan document. 

If the document allows X dollars per night of out of town stay, then up to X dollar would be non-taxable fringe benefits for the employee.  X would be capped at $50, because the Regs allow only $50. 

If the document allows reimbursement of airfare, it'd be nontaxable fringe benefits there again.  (Likely subject to the "reasonableness test" - e.g. NO first class to Japan even if other requirements for a medical trip to Japan are satisfied.  Likewise, NO flight by the Concorde to Paris etc. )

If the document allows Y cents per mile for driving, again nontaxable fringe benefits, with Y capped at the applicable IRS medical mileage.   

Now, do enlighten me about the "religious and political War".