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.
There are various parts to this; I'm focusing on only the overall meaning of each (as opposed to, say, the nuance introduced by "just"). So, I can construct a scenario where Example 1 is perfectly correct, but it means something different from Example 2. I'll use a slightly different example to illustrate:
Example 3
My friend: I am going to draw various things. For each drawing, I'll tell you I have done it and I want you to draw the same and then tell me when you're done.
Me: OK
My friend: [scribble scribble scribble] I have drawn a set of keys sitting on a desk
Me: [scribble scribble scribble] OK, I have just drawn the same thing
My friend: [continues onto next drawing]
Example 4
My friend: [Out of the blue holds up a picture he's drawn] Look! I have drawn a set of keys sitting on a desk
Me: Holy crap, that's amazing! I just drew the same thing!
My friend: [stares at my drawing, then me, then back at my drawing, terrified, as if I might be some kind of witch]
It wouldn't be too hard to morph that into something to do with thoughts rather than drawings -- maybe a scenario with a stage magician/mind-reader asking people to think of things and then tell him when they've done it. Doxastic involuntarism aside, I think the comparison works.
But, ignoring that contrived example, with the meaning implied in your context I would not use Example 1. Example 2 is better/idiomatic.
Best Answer
Out of other context (other related verbs), the meaning of the three sentences is the same. Something happened in the past and the result is now available.
As a possible improvement, the meaning would be more clear if you say: