Learn English – Is it correct to use the past simple with ‘Since’

The tense names are helpful in answering this question. The past tense in:

He lived here next door to me for three years.

indicates that he no longer lives here.

The present tense (perfect simple / perfect continuous) in the other two sentences:

He has lived here next door to me for three years.

He has been living here next door to me for three years.

indicates that he is still living here.

Note: it is just possible to imagine a context where He has lived here next door to me for three years (present perfect simple) does not mean that he is still living next door. For example, you could be listing all the places he has lived in his life:

He's lived for two years in New York; he's lived in an apartment in Paris for 6 months; he's lived here next door to me for three years; and now he's living in the Australian outback.

... you can't present perfect (or continuous) and past simple within a sentence.

As it stands, this rule is incorrect. In many cases it is acceptable and logical to mix past and present references in consecutive clauses

I lost my keys last week, but now I have found them.

This makes sense: A was true then, but B is true now.

This, however, does not make sense:

He has decided to go hiking, so I went hiking as well.

This sentence amounts to A was true then, because B is true now. The simple past describes a past event, your going hiking, but what the present perfect describes is not a past event, his decision, but a present state which is the result of a past event--his state of having decided. That present state cannot be the cause of the past event. The cause must be either a past event or a past state:

He decided (event) to go hiking, so I went hiking as well or
He had decided (state) to go hiking, so I went hiking as well.

The important thing is not to mix time references illogically.

As for the sentence in your friend's email:

Getting that email was such a pleasant surprise, because I was just thinking how I've been wanting to send you an email

There is no mixture of time references here, because the progressive construction "I have been wanting" marks a state, not an event, which may very reasonably be taken to continue into the present out of a past which is marked (by "just") as immediate. In effect, these pasts inhabit the same time frame as the present.

In any case, the "rules" are very loosely applied in informal discourse; see my discussion here. A casual email, which your friend probably dashed off in excitement, should not be held to the formal literary standards of coherence.