to include first single word in bigram or not?
What you describe is called padding and is indeed used frequently in language modeling. For instance if one represents the sequence "A B C" with trigrams:

# # A
# A B
A B C
B C #
C # #


The advantages of padding:


it makes every word/symbol appear the same number of times whether it appears in the middle of the sequence or not.
it marks the beginning and end of a sentence/text, so that the model can represent the probability to start/end with a particular word.