There are some differences between ni and tampoco. An important one is:
No puedo ni mirarte después de lo que hiciste = I can't even look at you after what you did.
Ni mirarte puedo después de lo que hiciste = I can't even look at you after what you did.
Ni lo habíamos considerado = We hadn't even considered it.
Tampoco lo habíamos considerado = We hadn't considered it either.
Tampoco can be applied to either Noun Phrases or Verb Phrases: in Juan tampoco compró las entradas, the most common interpretation is that Juan didn't do several things, among which was buying the tickets. That is:
[Juan] [tampoco compró las entradas] ⇒ Tampoco applies to the VP compró las entradas
In Tampoco Juan compró las entradas, one can deduce that there were several people that didn't buy the tickets, among which was Juan. That is:
[Tampoco Juan] [compró las entradas] ⇒ Tampoco applies to the NP Juan
The same goes for ni: in Juan ni compró las entradas, Juan didn't do several things, not even buying the tickets. In Ni Juan compró las entradas, several people didn't buy the tickets, not even Juan.
Now, ni has other meanings besides (not) even. It can also mean (n)either, or (n)either... (n)or... when it's duplicated:
No llamará (ni) el lunes ni el martes = [S]he will call neither on Monday nor on Tuesday.
Ni Juan ni Sofía van a venir mañana = Neither Juan nor Sofía are coming tomorrow.
What about Ni yo/Yo tampoco?
In both cases ni/tampoco can only apply to a Noun Phrase, so the only possible deduction is that there were other people who didn't do something besides you.
A: No he ido al centro esta semana.
B: Ni yo / Yo tampoco.
Here, A hasn't gone downtown, and neither has B, and both sentences mean the same thing. There’s a third option: ni yo tampoco, which combines the previous two.
How does “Ni tampoco” work?
“Ni tampoco” is used in negative polarity environments:
Ni quiero que me ayudes, ni tampoco quiero tus consejos.
“Ni tampoco” can’t be used to mean “not even”, and is more emphatic than just “tampoco”. “Ni tampoco” is the negative counterpart of “y también”, and as such it is used as the second term of an enumeration; if there’s only one item you can say “no… tampoco”:
No quiero que me ayudes. No quiero tus consejos tampoco.
Some grammarians consider that “tampoco” must agree anaphorically with the polarity of the verb, so to speak: that is, since “tampoco” is negative, the verb must be in negative too. That makes both of the following sentences incorrect:
No quiero que me ayudes. Quiero tus consejos tampoco. [“quiero” is affirmative and “tampoco” is negative so they disagree].
No quiero que me ayudes. No tampoco quiero tus consejos. [“Tampoco” already makes the verb negative, so “no” is redundant].
The interesting thing is that, whilst “no tampoco quiero tus consejos” is incorrect, “ni tampoco quiero tus consejos” is not. Arguably, the “no” component of “ni” is redundant, so “y tampoco” should be the correct option, but for some reason both options are okay, provided you don’t use a “ni” before:
No quiero que me ayudes, y tampoco quiero tus consejos.
So a fourth option added to the three you already mentioned is:
Y yo tampoco.
Now, “ni yo tampoco” is rare, and “y yo tampoco” even rarer, so I don’t recommend using the first one except when you want to be emphatic, and just never use the second.
NOTE: some example sentences were taken from the NGLE.
