All of your examples show continuous actions in the past that were interrupted by short events in the past. In such case you use the past continuous (progressive) tense (for the ongoing action) + past simple tense (for the interrupting action).
The past continuous (also called past progressive) is a verb tense
which is used to show that an ongoing past action was happening at a
specific moment of interruption, or that two ongoing actions were
happening at the same time. (Englishpage.com).
So, only this example is correct:
(3) I was tricked a few times while I was shopping at ABC.
About your thoughts:
Most of us think "never" and "ever" allow a tense shift because these
words indicate an event happening in the past, leading up to the
Both never and ever can be used in many tenses in the past, present and future.
So, it is okay to have mixed tenses (present perfect and simple past)
in (1) and (4).
In the sentence 1: I have never been tricked while I was shopping at ABC. the first part (have been tricked) is in present perfect progressive, and the second part (was shopping) in in past progressive.
In the sentence 4: Have you ever seen a tornado while you were driving to work?, there is also present perfect progressive, and past progressive.
These tenses do not go together, because, when the clauses in each sentence are connected by "while," past continuous can go only with past continuous or with past simple.
(3) is certainly correct because the simple past is used in the whole
The sentence: I was tricked a few times while I was shopping at ABC. is correct and the tenses used are past simple and past progressive.
However, the mismatched tenses in (2) make it grammatically wrong
The sentence 2: I have been tricked a few times while I was shopping at ABC. uses the same tenses as the sentence 1 I have never been tricked while I was shopping at ABC. and they are both wrong as explained above.
and the present perfect in the dependent clause of (5) "while I have
been..." makes it sound odd.
The sentence 5: Have you ever seen a tornado while you have been driving to work? uses present perfect simple in the first part and present perfect continuous in the second part and I don't think these tenses can go together when connected with "while."
In conclusion: When a sentence contains a clause in past progressive that starts with "while," the other clause can be only in past progressive or past simple tense. (Check: EnglishPage, Education First). But you can use "never" or "ever" in sentences with mixed tenses, not only in the past, but also in the present and future tenses.