1) Shadowmere will stay where you leave him if you mount another horse.
2) I think Frost behaves the same way.
That gives you two horses that will stay at your home as long as you mount another horse before fast travelling.
What about a horse that you own other than the previous two? If I buy a horse in Riften and bring him to Lakeview then mount another horse (arvaak) will that horse stay there or will he start heading back to Riften?