schaden = schädlich, nachteilig sein, eine Beeinträchtigung, einen
Nachteil, Verlust darstellen; Schaden zufügen (Duden)
"Schaden" is the most general one and it means to have a disadvantage or negative effect. It can be used for objects and persons.
schädigen = bei jemandem/etwas einen Schaden hervorrufen (Duden)
"Schädigen" is often used if its not a physical damage (scatching the object/person), but more a general damage to a state or situation, e.g. "jemanden finanziell schädigen" (=ruining oneone financially) or "jemandes Ruf schädigen" (=ruining a reputation)
beschädigen = Schaden an etwas verursachen; schadhaft machen
(Duden)
You can only "beschädigen" something not somebody, e.g. "fremdes Eigentum beschädigen" (=damage someones posession)
In regard of your examples:
Rauchen schadet dir.
Yes, that's correct.
Rauchen beschädigt dich.
No, you wouldn't say that as "beschädigen" refers to objects.
Das schadet der Natur.
Yes, that's correct.
Das beschädigt die
Natur.
Reading this sentence, I wonder "who does it damage nature". As the damage is less specific I would say "Das schadet der Natur".
Sein Verhalten schadet unserer Freundschaft.
Yes, that's correct.
Sein Verhalten beschädigt unsere Freundschaft.
Yes, that's correct. However, as a native , I prefer schadet (I can't give an explanation why).