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.
Short answer: No. "Some day" refers to the future.
I think most English speakers would say that (c) is just wrong. It doesn't make sense, precisely because it breaks the "future" rule of the idiom.
(a) and (b) are acceptable, but that's because they are not really using the idiom, they just looks like the idiom. You're pulling two words out of the middle of a sentence that match an idiom, but it's not really being used the same way. If that's not clear, consider the phrase, "Something came up." This is a common idiom that means, "some event happened or is expected to happen that interferes with previous plans". Like, "I'm sorry, I was planning to attend your party, but something came up." Now consider the sentence, "We forced air in to the bottom of the well, and something came up the pipe." That includes the words "something came up", but clearly we are not using those words in the idiomatic sense. Here we mean them literally, and it is just a coincidence that they look like the idiom. Same thing with sentences (a) and (b). You're not saying, "I saw him some day" in the same sense that you would say "I will see him again some day", and then clarifying that that day was last month. Rather, you literally mean that there was a particular day last month, some day, when you saw him. Similarly for (a).
As JR says, (a) sounds awkward because it looks too close to the idiom, and thus is confusing. I think most English speakers would say "some time around 1900".
Best Answer
You wouldn't use the past perfect with that idiom in the past tense. You want to make sure the tenses match, "He had enough problems as it was."
It's slightly awkward phrasing, though. Remember just because you can legally make a sentence in English (or of course any language), doesn't mean it's the best way to say what you want to say.
It's hard out of context to say exactly how to say it better, but for example, "John didn't want to start a fight with Maria -- he had enough problems (as it was) and didn't need to add another." Here "as it was" is fine, and if that sounds natural to you then roll with it. Personally, I'd write something like "he didn't need to add to his current list of problems" ... but, the more I think about it, the more "as it was" seems fine.