1) I had not really considered these points before, but I almost completely agree with you. I think keeping Knowledge, Light, Trickery and Nature in light armor, Life (maybe) and Tempest getting medium, and War getting Heavy would have made a lot of sense. Martial weapons for many of them would be fine (it irks me when I can't take a warhammer as a weapon).
The martial weapon proficiency bothers me a lot less than the heavy armor. I'd still rather see something like "simple weapons, plus one martial weapon". I know that most PCs would really ever use one or two martial weapons, anyway, so there's minimal in-play difference. It's just much more thematic. The Tempest and War Cleric may both only ever care about having a warhammer to use, but the fact that the War Cleric
could pick up a greatsword, whip, maul, etc. makes it
feel different.
Which is actually one of the reasons I don't like heavy armor proficiency being as common. It cheapens the flavor of the War Cleric if half of the domains grant that proficiency. The Tempest Cleric can still have heavy armor, if the table is using feats. Yes, that means paying for something that's currently free. Give them a different ability that's still balanced, but more flavor-appropriate. I'm not trying to de-power the class, just make it less samey-samey. The ubiquity of heavy armor for Clerics just feels like the developers phoning it in.
The Life domain is a special case. I really hate it granting heavy armor. Not just because it's samey, but because it feels like it runs totally counter to what the domain
should do.
2) I also mostly agree with this. However, I would suggest each domain getting two channel divinity options, some of which will be turn undead or an effect similar to turn undead. I do wish they could channel divinity a number of times equal to their Wisdom modifier instead of a set number of times a day, however.
I would be totally fine with effects that are similar to turn undead, but different. Maybe the Tempest gets to turn elementals. Nature might turn abominations. Knowledge gets to turn imbeciles (good in an election year). The point isn't that I hate turn undead. Just that, once you lose the pseudo-Christian context, it makes little thematic sense. I
definitely want channel divinity to be a relevant ability.
As far as Wisdom bonus vs. set number of times, that feels like a balance thing. The Cleric class has enough thematic issues that I can't even make it to the point of evaluating the balance. Really, balance generally comes down to numbers. Give the class some flavor, then adjust the numbers until it works.
3) This one I kinda disagree with though. I think their damaging spells is pretty limited. Or maybe the spells they do have are enough, but are a little lackluster or badly designed (ex. Sacred Flame should be roll to hit, not a save). I think they get some pretty good added domain spells, although I think some could be switched out for better choices. What I wish they did have was some more, unique to cleric control or influential spells. Not necessarily things that do damage, but ways to influence the stage or environment around them.
Actually, as I sit here writing this, I think I would have been fine with a much smaller Cleric spell list, and much more expansive domain spell list for each domain (say, 5 domain spells per spell level, going all the way through 9th level spells). Make each domain much more unique. For example, you'd end up with all clerics still getting basic healing spells like Cure Wounds, but only the life cleric with access to more powerful ones like Mass Cure Wounds, Heal, or Aura of Vitality.
I actually couldn't name a ton of damaging spells for Clerics. It might just be the design of giving damaging cantrips to every class that bugs me. It could also be the standardizing of the damage done by spells of a given level. Clerics have always had some damaging spells, but I recall them as generally being either a higher level or smaller/fewer dice than the closest equivalent Wizard spell.
I think I like your idea of smaller base spell list and more domain spells. It actually makes sense for the Tempest to get
lightning bolt and do it on par with the Wizard. War might do as much overall damage, but it'd be more in the form of
magic weapon and the like. Knowledge should have access to more divination spells than others. But, the same thing could be applied to Wizard schools -- which will be a topic for much later.
Really, the core issue is that, over the editions, they've standardized the spell preparation math and the effects of spells by level and probably other factors, but they've left the Cleric with access to the full list of spells while the Wizard (and other arcane casters) are stuck with the restrictions they've always had. In AD&D, the DM was explicitly supposed to review the Cleric's list every day and potentially swap out spells they didn't like. Players who started with 3E or later would
freak at that idea, but it was a very real balancing factor.
Want to have some fun? Get a deck of spell cards. Have the player select 1/3 to 1/2 half their spells every day, as normal. Shuffle the deck and deal out the remainder. It's not really the DM doing it, but it sure feels like the "will of the gods". Domain spells are still "gimmes".