The disadvantages are the same as wearing regular armor you lack proficiency with
(Note, I used the basic rules pg.44 as a reference, but the information should be the same as in the PHB, pg. 144)
You claim that shields are not armor. I challenge this assertion. Shields are not body armor (as noted in this answer on the first linked question). The answer on the first linked question is pointing out that all body armor changes how AC is calculated, while shields give a flat +2 AC.
However, shields are still listed in the armor table along with all light, medium, and heavy armor. The section title that you quote is "Armor and Shields" and the section begins:
Anyone can put on a suit of armor
or strap a shield to an arm. Only those proficient in the
armor’s use know how to wear it effectively, however.
Additionally, shield proficiency is noted in the "armor proficiences" section of the class description.
Based on the fact that shields are lumped together when talking about armor in the armor table itself, in the beginning of the preceding section, as well as in the armor proficiencies of the class description, shields are meant to be considered armor. Thus, whatever disadvantages come from using armor that you are not proficient with also apply to using shields if you lack proficiency.
As an aside and a note on your second linked question, just because a shield does not interfere with the Draconic Resilience of the sorcerer, doesn't mean that the sorcerer doesn't need to get a shield proficiency from somewhere else, like multiclassing, to use it effectively.
Yes, this works
You have quoted all of the relevant bits, but the Animated Shield works the same as a wielded shield without Wielding it with your actual hands that are holding onto a Heavy Weapon.
This is also a Very Rare item - it's special and powerful!
Sage Advice confirms
The key phrase comes from the Animated Shield, which states, emphasis mine (DMG, 151) :
...protects you as if you are wielding it
"As If you are wielding it" is equivalent to "wielding it".
This interpretation is supported by Jeremy Crawford's Sage Advice.
The text of animated shield says the item protects you as if you were wielding it. To Shield Master and the like, you're wielding it.
Not without a cost
The investment cost is two of your ASIs (and you still need to get, and keep, that Animated Shield.)
(Don't?) Hold on a minute...
Just a reminder, but the Animation requires a Bonus Action and only lasts 1 minute. You only get ONE bonus action/turn so your first round would be used doing this (so no, extra attack for a crit/kill from GWM or any other Bonus Actions you may have available.) In addition, if your encounter lasts longer than 1 minute, the shield will fall unless you use another Bonus Action to start it back up.
Best Answer
No (but your DM can allow it)
Player's Handbook, p.144:
It doesn't say you only gain the AC bonus from one shield, it says you cannot benefit from more than one shield. If you attach a second shield, only one of your shields has effect (unless a more specific rule elsewhere overrides that).
Note that it doesn't say which of your two shields has effect, if you've attached two. You could, arguably, swap between them ambidextrously as benefits you, but there's no particular rule covering this rare case.
Dungeon Master's Guide, p.141:
This doesn't explicitly describe shields, so it doesn't override the general rule that you can only benefit from one shield at a time. The fact that it doesn't say "you can only use one shield" doesn't implicitly mean that you can use two shields, because the general rule for shields is that there can be only one.
However, it does advise the DM to "use common sense", which means that it's up to the DM whether or not you can actually wear and benefit from two shields.