I've noticed recently that Shadowmere is dying from attacks from much lesser beasts. First time was a frost dragon. I thought that was odd. Next, a bandit took him out. Today, a sabre cat killed him. Anyone else having this problem?

Not here ... here Shadowmere was kidnapped by a butterfly: http://www.gamesas.com/topic/1367653-shadowmere-was-kidnapped-by-a-butterfly-and-now-is-gone/

If you have ever by mistake used marked for death on him, the effect is permanent.

All horses in Skyrim have problems. They seem to think that they are incredible hero Nords and are getting killed because they never stay away from battle, they'll attack alone on dragons, run between enemy and your ongoing powerhit, they are never scared or run away, but they always want to face their strength with trolls and giants.