Yes, it is possible to use "ever since" with the present perfect. "Ever since" links two events, a cause event and something that is true from that point in time onward. This is actually ideal for the present perfect tense because one of the uses of the present perfect tense is to indicate that something which began in the past is still relevant in the present. An example of this use is your sentence:
"I always thought it was wrong, but ever since I read it, I've been more confused."
Using the present perfect with "ever since" shows that the speaker has been and is still confused by something which they read in the past.
With the past simple tense, as in your last sentence, it would be more appropriate to use "once" instead of "ever since". This is because "once" links a cause event with something that changed, but does not necessarily link to the present.
Once I did that, it stopped bothering me.
This sentence shows that the speaker was not bothered by "it" as soon as they did "that". It however gives not indication of whether this continues to be true in the present.
I saw your edit, if I understand right you meant to ask if it is possible to use "ever since" and present perfect without past simple, which would look something like:
"ever since" + present perfect + present
Ever since I have been pregnant I feel sick each morning.
The answer is no, this is not natural English. This structure may be encountered in informal English, but it is more natural to have the structure:
"ever since" + past simple + present perfect
Ever since I got pregnant I have felt sick each morning.
To your first question:
I have been going on package holidays for years and I have never had any problems.
The Present Perfect Progressive has the meaning of "lately" or "recently." There is no indication from the second phrase that she stopped going on these trips long ago in the past. In fact, it sounds like she still may be taking trips, thus the answer.
The answer to your second question also lies in the second phrase. He is referring to the fact that he was never sick before from the foods he had already tried, not the food he is continuing to try, because obviously the most recent food made him ill.
I have been exploring the world...
The distinction is slight, but perhaps can best be explained as, there is no indication that the subject will stop the activity. The alternative
I have explored the world...
does not hold that implication. It could easily end with
I have explored the world for over 15 years, and it's time for me to settle down.
Best Answer
for an action that began in the past and continues up to the present.