Yeh dummies can be great for them when they are teething. I myself wouldnt removing it but its totally up to you.
DS use to be the same. Would wake and cry when the dummy fell out. But i found once i put him to bed, if i removed the dummy while he was asleep, he was less likely to wake during the night when it fell out. It was the falling out part that would wake him. Maybe you could try that. Other times yeh i just pursisted with it. You may be able to put a dummy in each of her hands soon too so she'll be able to put it back in. Wont be too long.. think DS was around 6 months old putting it back in.

Re the ribbon or chain, for sure during the day when you can keep an eye on them, but not such a good idea at night as can cause stranglation.

Best of luck.