For a 5 year old I would go with something like a 100-150 cm bow. Maybe drawing 5lbs max with double nocks so the weight can be raised slightly by using the inner nocks seeing as the little guy would be growing fairly quick.
Once he outgrows the inner nocks you can simply pike it and cut in two more sets of nocks and the bow will live on.
I would also make a 3 strand braided fishing line braid the length of his intended draw and attach one end to the riser the other end to the bow string. This will make it impossible to pull that thing back just as far as he can and snap it. Kids are prone to this with wood bows, especially boys who at that age have more energy than common sense.

You can find safety tips for arrows. Youll need to find some really light shafts though. Cork might work but I assume it would break down fairly quick. Soft, thick rubber would be my first choice.