Full Disclosure - I'm not a fan of dream sequences. I can think of perhaps a handful of one-off dream sequences in books that I've enjoyed and felt added value. I can't think of a single recurring dream sequence that's improved a book I've read and these days I skim them at best. But this is only my preference/opinion - and I'm not arrogant enough to assume it's the only valid one. I'll do my best to avoid biasing the answer!
I'd strongly suggest avoiding as much actual repetition as possible - if the dreams themselves are largely the same with only minor variations you can describe it in detail the first time it occurs during the story and for subsequent occasions indicate that they had the dream again and then describe the ways it differed. In this case the fact that they are having the recurring dreams is far more important and useful than repeating the actual dream content in detail.
I can't help but feel that the dreams are a little pointless to the big picture of the plot. Yes they illustrate important things about the character and yes they setup a climax after the last dream
You've hit on a key issue here - outside of dreams of unnatural origin (magically or technologically induced) it's hard to make the content of a dream directly relevant to the plot happening in the waking world of the book. You can paint some broad strokes about the character's state of mind but given the whole minefield of "interpreting" dreams you're better off doing the bulk of the heavy lifting in the character's reaction to the dream than in the actual content. Even so I find it's something that is only effective in small doses.
if I hide clues into each dream that build on top of each other like a puzzle that slowly comes together by the end of the last dream
I'm not going to lie, I winced a little at reading that. If by "hiding clues" you mean "provide details that make sense later" that's one thing. If on the other hand you mean hiding, hiding them - seemingly inconsequential details/differences in a repeated dream sequence that you hope the astute reader will pick up on and put together. That seems like a recipe for disaster. If there's a repeating dream sequence then possibly as early as the second iteration, and quite likely by the third the reader will tuning out a little.
Even if they don't dislike the sequence human brains reduce neural activity when processing repeated stimuli. While you can use that to your advantage in generating a bigger reaction to a bait-and-switch moment when they're expecting a repetition it means that subtle differences are more likely to be missed.
If you're looking to use the dreams to build towards something for the climax I think you'd be better served by distinct rather than recurring dreams, it will keep the reader's attention better and it will let you "hide" which details are important better while still giving them a fighting chance to figure it out.
I don't feel though that dreams that build on top of each other this logically are very realistic?
Well it's not especially realistic, but fortunately fiction lets you treat realism as more of a guideline than an actual rule. Assuming a non-fantastical setting all you have to do is keep the knowledge underpinning these "clues" as something the character knows (or could realistically know unconsciously) and you're fine.