I was driving my way back from work.- I'd remove 'my way' or add 'on'
As I was approaching this T-intersection with traffic light, - I'd drop the bit about the lights- it sounds a bit funny to me and is it really necessary to mention them here. When they appear, we know you're at a junction, so the presence of lights doesn't come as a shock.
turned a bit into the sidewalk- I'd remove 'a bit'
til the light turns green- spelling- till- also, it should be turned
hand-gesturing- gesturing or gesturing with their hand
she got off the car - out of
approached to me- approached me
they don't need help- didn't
"I'm on signal"- I'm on a red light/ I waiting for the signal
1- already answered
2- it's what I would say
3/4- already answered