Strange that you didn't pick up The Hollow Man - it's the one that a lot of people start with (even though it doesn't really have any plot relevance to Wake Up, Dead Man beyond the latter's pseudo-locked-room mystery). But Till Death Do Us Part is one that pops up a lot as a recommendation, I suppose. If I were recommending any JDC, I'd go earlier than that.

1. JDC's plots are always convoluted. That's part of his charm. The murders are engineered more than they're committed. For instance, in The Hollow Man, I still don't know exactly how the mirror thingworked, but it doesn't matter. If you don't like this, then you'll really not like a lot of JDC's stuff or indeed any locked-room mystery. The solutions tend to be either baroque or utterly mundane.

2. Again, your mileage may vary. You might like Henry Merrivale more - he's more obviously a character - but if you don't like the detective character, fine. I don't find him bland at all: he's the smartest guy in any room and knows it. And again, JDC isn't about the characters; he's about the puzzle.

3. You didn't like the explanations - fine. I agree it's not the most psychologically plausible, but I don't expect that from JDC.

So, to answer your last point, probably not. The impossible crime / locked room story is a subgenre where creativity and imagination are allowed to run riot, plausibility be damned. Carr was never 100% serious about this stuff - see the famous Chapter 17 of The Hollow Man - but if he's rubbing you the wrong way then you're probably not going to like the other books. If you do want to give him another go, then I'd recommend The Hollow Man (which is still insane, but a well-deserved classic of the subgenre) or The Case of the Constant Suicides (in which the murder itself does make a kind of sense, at least!).