Learn English – My friend neither/Neither has the friend

^{(I suspect this is not actually a case of double-negation — nor one of negative-concord, either. Maybe.)}

It might sound a little bit old-fashioned or formal, but yes, your second sentence is certainly allowed. See here. I might well use your sentence, at least in writing or formal, crafted speech.

It’s a pernicious myth that nor cannot be used without having been preceded by *neither. It’s up to the writer and their preferred style for that sentence, not some imagined rule grammar:

1. Those on the bottom had neither the ability nor the opportunity to get to the top.
2. Those on the bottom did not have the ability, nor the opportunity, to get to the top.
3. Those on the bottom did not have the ability or the opportunity to get to the top.
4. Those on the bottom did not have the ability or opportunity to get to the top.
5. Those on the bottom hadn’t the ability or opportunity to get to the top.
6. Those on the bottom had no ability nor opportunity to get to the top.
7. Those on the bottom had no ability to get to the top, nor the opportunity.
8. Those on the bottom had no ability to get to the top, nor had they any opportunity to do so.

I have seen broken software try to tell you that only one or two of those are allowed, but that’s just a bug in the limited experience with real English by the misdesigner of that software.

Using "neither" can be tricky, so in general, you can stick with "either" and never use "neither". Even native speakers frequently use it a way that goes against formal grammar rules.

Your first example is fine, and again you can stick with using "either" this way, even if the context is negative.

For lunch I'm either going to have spaghetti, or I'm not going to eat at all.

As in your second example, the "standard" way ensure you use neither correctly is to pair it with "nor". Just like you would use "either ... or" to create a list or choice, you can use "neither ... nor" to create a negative list (or lack of choice).

For lunch I'm going to have either spaghetti or a sandwich.

For lunch I'm going to have neither spaghetti nor a sandwich.

Your third sentence is an example of vernacular -- this construction shows up in various dialects, but it's not "formal" English and so I would recommend against using it. However, the only thing wrong with it is where you place the "neither", which should be in the middle, and you have to reorder the subject/verb pair:

I am not going to hurt you, but neither am I not going to protect you.

I don't know if there is any actual rule for this. It may just be another English structure that you can memorize and use as needed. Other examples:

I'm not going to have spaghetti for lunch, but neither will I have a sandwich.

She isn't at home, but neither is she out on a date, so I don't know where she is.

It's New Years Eve! I haven't been invited to any parties, but neither am I just going to stay home. I think I'll go find a nice crowded bar and celebrate there.