My horse does the same thing!
I slather his nose with desitin in the mornings- it keeps the sun from burning his lovely pink nose and also prevents it from being chapped by the dew.
As for the flymask issue…
I also use a Cashel long nose with ears, which my horse rubs off, shakes off, and removes magically whenever my back is turned. I have found that I can keep his mask on by pulling a piece of his forelock back over the headstall-ish part of the flymask and braiding it into his mane. I secure the whole thing with a rubber band and it stays really well.
Hope this helps!