There is no problem starting a sentence with the word and. Though we would not
necessarily choose to start a short sentence with and, it is not incorrect. When many sentences are strung together with and between them, you must call a halt and start a
sentence with and. The rule of thumb is no more than three separate sentences before you
use a period.
…She was walking near the tracks, and she did not seem to be having any problems, and
she noticed the sound of a train. And she didn’t pay a lot of attention, and then she began
to walk right on the track. And she felt a vibration in the tracks, and she knew a train was coming. And she still didn’t seem to react to it, and she kept walking.