Can wizards wear cloaks or capes instead of robes? I know armor is not allowed because it prevents the movement needed to perform spells but a cloak or a cape is not specified in the PHB, at least not that I've noticed.
There are no rules that allow this, but the DM might allow you to loot a defeated suit or wear an active one with roleplaying consequences.
Note: in 5e the Helmed Horror seems to be a superior, but mostly identical, version of the Animated Armor as referenced by name in the Helmed Horror's entry (MM p.183). After defeat the armor MIGHT be recoverable, as per MM p.15:
You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Player's Handbook for inspiration, and you decide how much of a monster's equipment is recoverable after the creature is slain and whether any of that equipment is still usable. A battered suit of armor made for a monster is rarely usable by someone else, for instance.
So it's the DM's call here. Furthermore, upon its "death" a suit of Animated Armor becuse a suit of Inanimate Armor, aka normal armor. So you won't be wearing a suit of Animated Armor per se. Another thing to keep in mind is that a suit of Animated Armor is made out of a suit of full plate armor (as per MM p.19). This costs 1500 gp, which is very high for treasure befitting the treasure you can find for that monster's challenge rating. So the DM should keep that in mind when creating treasure for that level.
Can I wear a suit of Animated Armor?
A suit of Animated Armor is a monster, not an item. Being inside of a monster is frequently the result of being the victim of Swallow Whole, which is not a good thing. And while the Animated Armor is described as being an "empty steel shell" in its entry, the Helmed Horror has got some magical red energy going on. Neither entry describes if you can remove the armor either. While there have been monsters in the history of D&D that improve a host's capabilities when bonding with them, the Animated Armor of 5e is not one of them. So by RAW the answer is No.
However, a player wearing a suit of animated armor might prove to be a fantastic roleplaying oppertunity. Given that the suit and the wearer are not at odds, having a conflict between the will of the user and the orders of the suit can make for some great roleplaying. Perhaps the wearer of the suit is a condemned criminal who was sentenced to wear the suit under the orders of a wizard lord and has to make amends for his crimes? Maybe the suit was send out to find someone or something, and a hapless adventurer put it on and now the wearer has to carry out some objective that does not clash all that much with the suit? This can all be really fun, but you will have to talk this over with your DM (preferably out of game) before you do this.
I think you're bypassing the "don" and "doff" rule for shields.
I agree that there's daylight to be seen between carrying and "wielding" a shield. But I'll pose you this question: if you can have a shield and switch states from "wearing" to "wielding" it freely, then what is donning or doffing?
As for the comparison to equipping a weapon, I don't think it's valid. No weapon classes are described as requiring an equip-time; all armor are described as needing time to make usable. (I'm ignoring the little bit of time-constraint provided by "loading".)
My read of your description above is that you've house-ruled the donning and doffing of a shield to cost zero in the action economy. If that works for your group, that's great. But I think it is a departure from PHB rules.