I allow lovies anywhere, anytime. In your case I'd mention to Mom that it needs a wash and mending "so it doesn't get more ripped." I've also run yucky blankets through a quick wash so I could stand them.
Usually by 2 I'm phasing pacifiers out by allowing them at nap only, then eventually they get "lost." But it depends on the child. I have a child who was very oral and not ready to start giving up her Binkie until she was almost 2 1/2.