You can find examples of most of these (thus demonstrating the proper way to use them) in the Sage Advice Compendium.
Class feature names
Class features, such as Cunning Action, Wild Shape, Draconic Resilience, Unarmored Defense, Empowered Evocation, etc., should be capitalized (but not otherwise italicized or anything).
Did we change certain things, such as Empowered Evocation, because they were overpowered?
For example, if you’re a sorcerer/monk, you can use either Unarmored Defense or Draconic Resilience, not both. Similarly, a druid/barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense (you aren’t considered to be wearing armor when you use natural armor).
Yep, they're capitalized. This can be easily found by looking at the Feats section of the Sage Advice Compendium:
The Charger feat relies on you, not your mount, taking the Dash action.
Elemental Adept is the only feat in the Player’s Handbook that you can take more than once.
Class feature subchoices (e.g. Eldritch Invocation names or Fighting Styles)
Yes, they're also capitalized. An example for Fighting Styles:
You can benefit from more than one Fighting Style option at a time, as long as they don’t have conflicting requirements, as Dueling and Great Weapon Fighting do.
And for Eldritch Invocations:
Eyes of the Rune Keeper lets you read any form of writing, including the linguistic meaning of a rune, if any.
Class and subclass/archetype names
Class names are not capitalized:
Like other spellcasters, the ranger follows the rule on components in chapter 10 of the Player’s Handbook. A ranger typically uses a component pouch for the material components of spells, but doesn’t start with one because rangers don’t have spells at 1st level.
A surprised rogue can use Evasion, since that feature doesn’t require the rogue to take an action or move.
But subclass names generally are capitalized:
Can a Circle of the Moon druid speak the languages it knows while in the form of an elemental?
That said, a DM would break nothing in the system by allowing an Eldritch Knight to reverse the order of the cantrip and the weapon attack.
Does a grapple or a shove trigger the Tempest cleric’s Wrath of the Storm or a Battle Master’s Riposte?
Note that there is a question where "thief" isn't capitalized:
Can a thief use the Fast Hands feature to activate a magic item?
...And several questions about wizards that refer to them as "[school]-er" instead of "[School of X] wizard":
The ward regains hit points whenever the abjurer casts an abjuration spell of 1st level or higher, not just when the ward has 0 hit points.
Does the evoker’s Potent Cantrip feature apply to cantrips with attack rolls or only to saves?
Does the 14th-level illusionist ability let you make something real every round, or just once?
That said, though these are referenced in the descriptions of at least some of these subclasses (e.g. the School of Abjuration wizard's), these seem more like in-universe terms used to describe members of the subclass rather than intentional stylings of the official/mechanical name of the subclass.
Abbreviations of ability names (e.g. Dex vs. DEX, for Dexterity)
The Sage Advice Compendium writes out "Dexterity" every time it's used... But the abbreviation does appear elsewhere in the rules text - specifically the Armor and Shields table in the basic rules and in the PHB (p. 145). The AC of leather armor, for instance, is listed in the table as:
11 + Dex modifier
And the Strength requirement for chain mail is listed as:
...So the correct format of abbreviating an attribute score seems to be to take just the first 3 letters of the ability and capitalize the first letter.
The abilities themselves (Strength, Dexterity, etc.) always have the first letter capitalized:
The shadow’s Strength reduction isn’t damage, because it has no effect on the target’s hit points.
Spells, and a note on magic items
Your understanding is correct. This is true for both spells and magic items; they're in all lowercase, except for proper nouns (and unless it's at the beginning of a sentence or something), and they're always italicized:
[...] Spells sometimes modify AC as well. Shield of faith, for example, grants a target a +2 bonus to AC until the spell ends.
Magic items can also enhance your AC. Here are a few examples: +1 chain mail gives you an AC of 17, a ring of protection gives you a +1 bonus to AC no matter what you’re wearing, and bracers of defense grant you a +2 bonus to AC if you’re not wearing armor or using a shield.
The attack from War Magic is made normally, and you get the benefit of true strike on your next turn if the spell hasn’t ended.
Similarly, if you bond with a dagger of venom, you can’t summon it as a maul; it’s always a dagger.
An example of one with a proper noun:
You currently have 2 temporary hit points remaining from armor of Agathys, you just slew a monster, and your Dark One’s Blessing can now give you 4 temporary hit points. If you take those temporary hit points, they replace the ones from armor of Agathys and end that spell, so you might not want to take them and keep the spell going instead, so as to continue benefiting from the cold damage it delivers.
Failing a save against Tasha’s hideous laughter means you’re incapacitated even if you can’t fall prone.
Other stuff you didn't mention
The names of specific actions are capitalized: Dash, Dodge, Use an Object, Cast a Spell, etc. Example:
If you can take the Dash action as a bonus action, nothing in the rules prevents you from taking the Dash action with your regular action too.
However, as you can see from the above quote, the classifications of "actions" within the action economy (action, bonus action, and reaction) never have their names capitalized. Only the names of specific actions are capitalized.
This is also true of the names of monster traits and actions, though the monster names themselves are not usually capitalized (unless referring to a particular named individual, such as Strahd von Zarovich):
Yes. Wing Attack does not protect the dragon from opportunity attacks.
An earth elemental’s Earth Glide is meant to apply to itself only.
A vampire can target a different humanoid each time it uses its Charm action, potentially causing an entire group of enemies to become charmed if their saving throws aren’t successful.
Correspondingly, the names of player races aren't capitalized either, though racial traits are:
If an elf meditates during a long rest (as described in the Trance trait), the elf finishes the rest after only 4 hours.
The lightfoot halfling and wood elf traits—Naturally Stealthy and Mask of the Wild—do allow members of those subraces to try to hide in their special circumstances even when observers are nearby.
Creature types (which both PCs and monsters/NPCs have) aren't capitalized either:
A tiefling is a humanoid, not a fiend, and therefore escapes the notice of Divine Sense. The feature detects creatures that have the celestial, fiend, or undead creature type.
Magic items are lowercase and italicized, as spells are (as mentioned above):
An instrument of the bards provides this benefit only if it is used as a spellcasting focus in the casting of a spell that causes its target to become charmed.
A spell scroll can be named in a variety of ways: spell scroll, scroll of X (where X is the name of a spell), or spell scroll of X (where X, again, is the name of a spell). No matter how its name appears, a spell scroll follows the same rule.
The names of languages (e.g. Common) are capitalized, which helps distinguish them from the name of the race that speaks it (which might be the same word but not capitalized):
The Monster Manual mentions other languages, including Gnoll, Modron, Sahuagin, and Slaad.
Creature sizes are always capitalized:
Can the monk’s Open Hand Technique push a Large or larger creature or knock it prone?
(More examples can be found in the section of the basic rules on creature size.)
And, of course, book names are always capitalized and italicized, but that's just a regular style guideline.
I also came across an official D&D style guide by Wizards of the Coast. There's an older version of the D&D Style Guide document (1.04a) here, but the latest version (1.08a) is available on DMsGuild as part of this free product: DMs Guild Creator Resource - Style Guide Resources. According to the product page, the style guide was last updated in January 2019 (this article summarizes the changes from 1.04a to 1.08a).
In general, it follows the Chicago Manual of Style and the Merriam-Webster's Collegiate Dictionary. It includes some styling details not discussed above. The specific official styling rules for D&D game terms are detailed on pages 6-7 of the PDF.
The package also contains this Forgotten Realms Style Guide (the version in the DMsGuild product seems to be unchanged from this PDF). It mostly provides setting details on tone and background as help for designers, but there are some guidelines provided on styling. For instance, it lists certain terms that aren't used in the Forgotten Realms (e.g. "o'clock"), certain terms and names invented for the setting (e.g. "peeptalk" for the language of birds), and the "correct" adjectival forms for citizens/products of certain locations (e.g. "Evenor" for people/things from Eveningstar).
(The Style Guide Resources product also includes a D&D IP Guide about important aspects of D&D's intellectual property, and a short guide by Chris Perkins & Greg Bilsland on the principles of writing adventures. Neither of these contains any styling guidelines.)