You cited a secondary meaning of everyday but the primary meaning is exactly the same as every day (with a space), according to Random House and World English dictionaries. Therefore these are synonyms.
Psycho has no meaning as a word except as a shortened, slang version of psychopath/psychopathic. However as a prefix psycho- means exactly the same as in the word psychotherapy. Actually it is the same.
Therapist is a combination of therapy and the suffix -ist. Therapy is derived from Neo-Latin but ultimately Greek. It has a different meaning because its origination came from very different words than the origination of the and rapist.
Let's look at another example - the word shock.
On the one hand the meaning of a violent blow or impact comes from the French word choc coming into the language in the 1500s. On the other hand the meaning of sheaves of grain, later also applied to hair, came into the language from the Low German word schok. So here you don't even need to add a space and you have two entirely different meanings with the same spelling. That's called a homograph. These words also sound exactly the same but have different meanings. So they are also homonyms.
You are adding a space so the spellings are not exactly the same (because of the space) so none of your examples can properly be homographs but they are can be homophones (with same sound but different spellings) or homonyms.
But in the example of therapist and the rapist these do not have the same spelling, sound, or meaning. The just have similar, but slightly different, spellings. Just like cold and cord, or cold and could.
Cold and cord are what is called orthographic neighbors or substitution neighbors. But I think your examples are not quite these because you have added spaces, and the relevant example is cold/could. These words are addition/deletion neighbors.
Any word that you add spaces to to create a phrase would cause the two to be addition/deletion neighbors.