Yep. And as a High Half Elf, you could pick up Booming Blade to help lessen the pain of the ridiculously late Extra Attack.
Indeed. I'd be more apt to go Life or Order for a Heavy Armor domain with this character. Either domain still represents "Paladin-y" values, while being better choices than War.Out of all the Cleric domains that give you Heavy Armor prof, War would be my last pick. It's just really lacking.
Another option would be to start Paladin 1/Cleric X, to represent them starting their journey as a Paladin without reaching the point of gaining most Paladin-specific abilities. Then you could go with a different Paladin-appropriate domain like Peace, Grave, or Light that doesn't natively get Heavy Armor proficiency.
Or, another option would be to stick to just medium armor Cleric domain levels, and take Heavily Armored instead of Heavy Armor Master. This would achieve the same result, with HA being a half-feat just like HAM, without you having to take the initial Paladin level.