All my kids start chores around 6 or so, but 5 is also a good age. I always start with putting their own laundry away. After I wash and fold it, they help carry it to their rooms and put them in the appropriate drawer. After that has worked for a while, I start with other things like helping with the trash, getting it out of the house & down to the curb. You will be amazed at how fast and how much a child can do and will do especially if they are getting rewarded. I did do money for chores in the begining, but with 4 of them, they are not paid any longer. Although, I see no problem with it, as long as they do the chores when told, correctly and with no complaints. Breaking any of these rules cost them the allowance. There must be a scense of responsibility that goes with the reward.
My kids range from 14 to 6 & here are some of their chores :
Putting clothes away
Trash
Vaccuuming
Dusting
Straightening pillows on the couch
Helping with picking weeds (We actually do this as a family every couple of weeks)
helping me put groceries away
Realy anything that can be taught simply and does not require a lot of coordination, I do not like to make my kids do anything I would hate to do (like clean restrooms or do floors). Picking up their rooms is a good thing for them to do, I only paid them for this, when they did it without me having to tell them and for more than 1 day. They had to show me commitment to this task as it is something I always considered their duty to do and not mine once they got to this age.
Good luck with the stealing, I have also gone through this with my 2 older children & I am not sure anything I did was the correct thing, but I was very annoyed by this behavior & am glad that at least for now, I have seen an end to that faze.