“She and I” is grammatically correct when it is placed before the verb and is the subject of the sentence.
“She and me ” are incorrect and should be “her and me” but only correct when “her and me” are the objects of the sentence.
She and I are going to the cinema.
The most important thing to remember is that “I” is the subject pronoun(the person that does the action) and “me” is the object pronoun(the person that receives the action. You can read more about the subject and object pronouns here.
When “she and me” is incorrect
“She and me” is incorrect at the beginning of the sentence because we do not use “me” as the subject of the sentence.
We are so used to subject(I) + verb that when this structure is disrupted, it can be easy to make a mistake.
I remember being corrected as a child that it is always “my friends and I “ and finding it difficult to say this structure.
I play tennis.
She loves me
she and I play tennis
When to use “she and me ”(her and me)
“Me” is an object pronoun that means that you receive the action. “Her” is the object pronoun of “she”
She punched me.
In this example, you received the punch. It doesn’t make sense to say “me punch”.
You can only use “her and me ” when you and the other person(she) receives the action.
They started a fight with her and me.
The artist painted a picture of her and me.
It is just a form of politeness to put “him” before “me”. In reality, it is probably better to use the person’s name as otherwise, it can sound awkward.
The artist painted a picture of Jane and me.