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.
What would happen if I used "not yet" in these sentences...
Both "not yet" and "still not" are fine, but they carry different nuances. "Not yet" implies, strongly, the notion of "but that we clearly expect to happen somewhere in the future." "Still not" is very much weaker in that respect. So each has the following sense:
1') I have not yet found what I'm looking for, but I'm determined I will.
versus
1) I still haven't found what I'm looking for, and I'm just about ready to give up trying.
Curiously, that difference is much reduced in your examples 3 and 4. Neither of those uses of "not yet" carries the same kind of inevitability that the phrase does in the earlier examples. In fact you could probably interchange "not yet" and "still not" with little or no change in meaning for points 3 and 4.
Best Answer
It's "save the date". "Safe the date" is a malapropism, like "for all-intensive purposes" (for all intents and purposes), "make my words" (mark my words), etc.