Anke, be very careful with your diagnosis, could cause a lot of panic and unnecssary expense
Andrea, for starters, the runny eye is no related to the cough, your vet will have a oil cream called Orbenin, it should help
Re the cough, a lamb will cough if it has drunk its milk really fast from a bucket/bottle, is it coughing between feeds?
Re the scabby mouth, are you sure its not dry milk from where its spilt from the teat?