Thank you for the replies! That makes sense to me.
@inafogg: I have not done so before no, but I tried tonight and the car wouldn't run. Same problem as the last bullet in my list of symptoms on the first post. Playing around with the turn switch (up, down, away, towards, twist.. you name it) nothing helped. After a bit of kicking, and I think I pulled the ignition cylinder towards me while turning the key, the car started.
However, I can't say for sure it was because of the turn signal that the car didn't start because the two times it has happened in the (very recent) past the turn signal was Off. I'll try later few more times maybe the car will be in a different mood.
Thank you again