To be honest, in my opinion, your dog really needs a further work-up. Did they do bloodwork at all? Chest xrays?
If her coughing is improving with Lasix, then you can let your vet know that. However, if she isn't taking it at all, then you won't know that. However, I wouldn't personally start a dog on Lasix without doing more of a work-up first, because we don't know what else may be going on with her. Also, I like to know whether kidney function is normal first and whether the pet is dehydrated, etc, before Lasix. And whether or not the vomiting of phlegm is related to the coughing or another problem, I don't know.
Also, Lasix is a diuretic, so if anything, it'll make her urinate more. Make sure she is drinking plenty of water. If she is holding her urine, try carrying her outside to her usual spot and see if she'll go there.
As far as throwing up her food, try feeding her very small amounts at a time more frequently than usual. Try a bland (1/3 chicken boiled 2/3 boiled white rice) diet for a couple of days. Also, don't let her guzzle a ton of water at one time. Small frequent amounts will stay down better.
Hope this helps and good luck!