It is interesting to add the times of the mail from Edinburgh to Falkirk, they help to appreciate the "E" time code.
Box Closes at 10:00am, Letters taken on paying one extra penny till 10:10am, Departure of Mail at 10:40am, by Railroad
Box Closes at 8:45pm, Letters taken on paying one extra penny till 9:15pm, Departure of Mail at 9:30pm, by Mailcoach
The two routes can be seen here:

The E
vening time-code tells us that the letter was sent with the second mail.
Now it is clear why the arrival date of the Falkirk postmark was on the next day.
If the letter was posted very late at Edinburgh the extra penny had to be paid by cash and we should see this postmark on the front of the letter:

But your letter was posted on time.
About the "C" stamper id I can only add that on June 1842 there were 8 stampers at the GPO of Edinburgh.
Their names, taken from Parliamentary Papers, were:

p.s.
James Sloan wrote a letter on November 1841 complaining that the salary as letter-carrier was £67 etc and as stamper it was £54 etc !