All your examples are grammatically and semantically correct, you can use both versions:
- Lava su mano: not as common as "se lava la mano"
- Se lava la ropa / Lavó su ropa: both versions are perfectly correct. The first one stresses in my opinion the fact that he washes his own clothes.
- Se olvidó los libros / Olvidó los libros: both versions are correct
But you are right : there are some verbs which are reflexive. There are even some verbs which have different meanings, whether you use them as reflexive verbs or not. For example:
- Llamar (ella llama a su madre / she calls her mom) , llamarse (ella se llama Pilar / her name is Pilar)
- Ir (él va al trabajo / he goes to work) , irse (se acaba de ir / he is just gone)
and many more, you can google for them or take a look here.
I am afraid you will have to learn by heart which verbs with which meaning are reflexive. There is no general rule for this in Spanish.
There is also a special case (or maybe the common case?) when someone is performing an action on themselves. In this case you use a reflexive construction like the examples you mentioned