How long does it take for one citizen to ask if he / she can take your dropped item? I need to do this if I want to be Thane of Falkreath... it seems like doing quests before being asked to assist people around town does not count.

I think it depends on what quests they are. I have done quests for people in a town and upon receiving the "assist the people of....." quest, it would show how many out of 3 or 5 I have completed already. In fact a couple of towns it was completed as soon as I got it and I was able to turn it in.
All the things which are considered helping: http://www.uesp.net/wiki/Skyrim:Thane_of_Falkreath

Drop a nice piece of clothes ( I had a woman pick up fine clothes off me before) or a nice piece of enchanted gear/weapon. They are more apt to go after that then lets say an old pair of boots.

I have always found that any help given before the thane quest is given has always been recognized in the thane quest.

so i start with 3/5 instead of 0/5 for the thane quest.

^This. They do count, Ive had it come up with 5/5 immediately.