"Nunca" and "jamás" can be interchangeable sometimes. But the nuance I find, is that "jamás", tends to sound a bit more dramatic and more "definitive" than "nunca".
"Tampoco" is used when you have made a negative statement and you're adding another one.
"Ni - ni" is a couple similar to "neither - nor".
"Ni" alone, stresses the absence of something.
Juan nunca me dice que me quiere. (She's sad) / Juan jamás me dice que me quiere. (She's about to cry or crying.)
Juan never says that he loves me.
Nunca le doy mi (número de) teléfono a extraños. (I'm a bit paranoid) / Jamás le doy mi teléfono a extraños. (I'm paranoid)
I never give my telephone number to strangers.
Nunca manejes cansado. (Wise advice) / Jamás manejes cansado. (More like an order.)
Never drive if you're tired.
Nosotros nunca comemos mariscos. (Simple statement) / Nosotros jamás comemos mariscos. (Clear aversion stated)
We never eat sea-food.
Juan nunca me dice que me quiere y yo tampoco le digo que lo quiero.
Juan never says he loves me and I never tell him that I love him.
Yo no le doy mi teléfono a nadie y tampoco hablo con extraños.
I don't give my telephone number to anyone and I don't talk to strangers either.
No me gustan los mariscos, pero tampoco la carne.
I don't like sea-food, but neither I like meat.
Fui a la farmacia, pero no pude comprar ni la medicina ni las jeringas.
I went to the drugstore, but could buy neither the medicine, nor the syringes.
¿No tienes ni café ni leche?
You don't have any coffee or milk?
Se nos hizo tarde y no fuimos ni al cine ni al teatro.
We were late and we went neither to the movies, nor to the theatre.
No tengo ni idea de cómo pudo pasar esto.
I have no idea on how this could happen.
Abrimos la llave pero no salía ni gota de agua.
We turned the faucet on but not a single drop of water was coming out of it.
Ni te imaginas lo bien que nos fue.
You can't even imagine how good we did.