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.
...I forget it now
...talks about that 'very moment'.
... I have forgotten
... the use of past participle makes it the 'process' starting from that point till now.
For a non-native speaker like me, the simpler version is...
ColleenV once told me the web site's name [sic] that I don't remember.
Best Answer
I don't know who told OP that "No. I have forgotten to" was a "wrong answer", but it was bad advice.
Since the "forgetting" is still "relevant to the current moment", Present Perfect is grammatically/logically acceptable in such contexts.
But it is true that in practice most native speakers would use Simple Past "No. I forgot to", because very often the actual action described using Present Perfect continues into the present (i.e. - when you say "I have lived there all my life", it always means you still live there).
The tricky thing about forgetting to do something is that effectively it means you forgot about it. But as soon as you admit to having forgotten about it later, you're no longer forgetting. Because it creates a bit of semantic/temporal confusion, natives speakers tend to avoid the implications of Present Perfect here.