I love the thematic elements of this subclass, but the mechanics could use a little work.

To be honest, I think the emphasis and mechanics around the Subconscious Realm are a bit over-specific for a general subclass. If you had a world or campaign that included the Subconscious Realm as part of the lore, it would be great, but it might be prohibitive for a GM to try to shoehorn that into their otherwise-established world. Having some general lore in your description is great, but it needs to be abstract and flexible enough to make sense in a variety of worlds. Again, if this is for a specific campaign, you're fine to be as specific as you'd like.

For your specific features:

• Subliminal Affinity - I would remove the Perception proficiency. This feature is already very strong, and Perception doesn't fit the theme of the subclass. Insight and a free 2nd level spell is both strong and thematic.

• Channel Divinity: Repose - I'm not sure when or where you'd use this feature. It's less beneficial than a long rest. Instead, consider making it only an hour long--in addition to the benefits of a short rest, grant the extra healing and removal of an exhaustion level. I'll also note that this is an odd use of Channel Divinity, which is typically something more combat oriented (though not always). This is a decent feature, but might be worth being its own ability, once per long rest.

• Partial Delve - like I said before, this is a really lore-defining feature that I can't see many GMs welcoming. I'm also not sure what this feature is supposed to do mechanically, as I can't find any information of what a character might do in the Subconscious Realm or why they'd want to be there, unless the GM starts fleshing that all out. And unless your whole party is this domain, it would either do nothing or your character would have a solo sidequest every long rest, which doesn't sound fun at a table. I'd consider replacing this feature (maybe move the short-rest version of Repose here instead).

• Memory Palace - as another commenter said, it's not flashy, but it's effective. Though keep in mind that other subclasses all get a damage boost at this level (either adding their Wisdom modifier to cantrips, or bonus damage to weapon attacks). It's not necessarily a bad thing to buck that trend, just keep it in mind for balancing.

• Dreamcaster - in addition to being lore-defining, this is quite powerful, even for 4th-tier play. If I'm understanding it correctly, the cleric could use this feature to put a bunch of enemies to sleep (though also themselves). And everyone takes a level of exhaustion when waking up. The mechanics feel complicated, and the aspect of the Subconscious Realm would need to be fleshed out to make more sense. I think there's potential in the save-or-take-exhaustion mechanic, but it could be simpler (e.g., "Choose any number of creatures you can see, who must make a Wisdom saving throw. On a failure, the creatures are subject to an intense weariness and take a level of exhaustion").

I get the impression you're really into the Subconscious Realm idea (I'm getting Tel'aran'rhiod vibes, which is cool). But the whole idea might be better suited for some sort of module (which might include some dream-related feats, spells, etc.) rather than a subclass. If you want to explore that more, consider making an adventure module or something in that vein for it.

Here are a few thoughts for features that might fit the overall vibe:

• While casting Sleep isn't terribly exciting on its own, why not play around with enhancing the Sleep spell? For example, a Channel Divinity option could involve increasing the number of hit points the Sleep spell effects, and/or at higher levels, targets put to sleep by your Sleep spell take psychic damage each round that doesn't wake them up? Extending the duration? Removing the V/S/M requirements? Lots of options here to make Sleep useful beyond tier-1 play.

• A feature that allows you to detect any (intelligent) sleeping creatures nearby. Situational, but could be a good additional feature for a level where you want to add a little more.

• Just before completing a long rest, the cleric can use the Sending spell without expending a spell slot, but the target of the spell must also be asleep or the spell fails.

Hope that's helpful!

First of all, I'm so happy that someone else is posting in this community! Welcome!

Since you asked for help with this class: I'm not convinced about it. It alternates between features that are borderline useless with others that are utterly broken.

Subliminal Affinity gives you proficiency in two of the best skills of the game and a 2nd level spell for free once a day. Despite that, it doesn't give you any other benefit in battle, so it's very dependant on the type of game that's being run - It will find great use in a social campaign, less so in a campaign focused on combat.

Repose and Partial Delve are barely classifiable as ribbon features. Repose allows you and your party to benefit from a worse long rest in 6 hours. I'm not sure which situation would require me to use this feature instead of spending 8 hours and do a long rest, as normal. Partial Delve is very flavourful, but mechanically speaking, it only allows you to gain the benefits of a long rest after 12 hours, which is worse than just doing a long rest, so I'm not sure what you're trying to achieve here. You could replace the entire shtick with a free cast of Dream, maybe?

Memory Palace is uneventful, but it's good. Charmed and Frightened are two of the most common status conditions in the game, but they aren't overly common. When it matters, you'll be glad to have this feature, but it's not going to break the game by any means.

Dreamcaster is bonkers. It's a save-or-suck effect that completely obliterates an encounter if you're allowed to pull it off. Forcing any number of creatures of your choice to be unconscious for 1 entire minute would allow your party to kill the affected creatures way before they can reroll the saving throw to end it. Sure, most enemies at that level have multiple legendary resistances, but I'm not sure if a feature should be balanced on the fact that it won't realistically land.
The mechanics are unclear: as written, the dream only ends after 72 hours or if you exit it, which means that taking damage outside of the dream (or, potentially, even dying) won't end it. As far as I can see it doesn't even require concentration, nor does it force you unconscious, so you can cast a spell beforehand and the use the feature to put everyone to sleep.
By comparison, Dominate Monster, which is an extremely powerful 8th level spell, requires (alongside an 8th level spell slot, of course) concentration and allows the target to repeat the save every time they take damage.