Here is an example of how this task can be done using Python:
# Initialize the list of quotes
twain_quotes = ['I have never let my schooling interfere with my education.',
'Get your facts first, and then you can distort them as much as you please.',
'If you tell the truth, you don\'t have to remember anything.',
'The secret of getting ahead is getting started.',
'Age is an issue of mind over matter. If you don\'t mind, it doesn\'t matter.']
# Print the initial list of quotes
print(twain_quotes)
# Sort the list of quotes
twain_quotes.sort()
# Print the sorted list of quotes
print(twain_quotes)
# Insert the new quote in the correct spot alphabetically
twain_quotes.append("Courage is resistance to fear, mastery of fear, not absence of fear.")
twain_quotes.sort()
# Print the list of quotes with the new quote inserted
print(twain_quotes)
# Remove the first quote in the list
del twain_quotes[0]
# Print the list of quotes with the first quote removed
print(twain_quotes)
Note that this is just one way to accomplish the task and there may be other ways to achieve the same result. Additionally, if you are using other programming languages, the syntax and functions may differ.