[RPG] Can wizards really cast spells in armor

Short Answer

No, you need 5 minutes to doff plate armor. You could use illusion or possibly transmutation to create the appearance of armor. To persuade your DM to have easy-off armor you might have to pay a real world or game world cost.

Long answer follows..

RAW

The RAW say that Heavy Armor takes 5 minutes to doff (PHB 146), so instant doffing not possible.

You also can't teleport out of it, because as you say you can't cast spells in it.

Illusion

It sounds like you don't actually want to use the armor as armor (you want to take it off as soon as combat begins), so why not just create it as an illusion? As an example, Disguise Self seems to do the trick. It's a first level spell too, so much 'cheaper' than teleport! If you're high enough level to be using seventh-level spell slots then your spell save DC is going to be quite tough to beat (in terms of others seeing through the illusion).

You make yourself — including your clothing, armor, weapons, and other belongings on your person —look different until the spell ends or until you use your action to dismiss it.

Even so it would take your first action of the combat to dismiss the illusion. However, you could use a Hat of Disguise (DMG 173), and in that case it would just be a case of taking off the hat, a free action:

While wearing this hat, you can use an action to cast the disguise self spell from it at will. The spell ends if the hat is removed. (DMG 173)

Transmutation

If you would prefer not to use an illusion, you could use a transmutation spell. For example, you could see if your DM would let you use Alter Self, though the wording of the spell is unclear as to if you can actually create the appearance of armor.

You transform your appearance. You decide what you look like, including your height, weight, facial features, sound of your voice, hair length, coloration, and distinguishing characteristics

"Clothing and armor" are not mentioned, so your DM would have to rule this one.

Persuading your DM

If you really want to go with armor which falls off quickly (which is a hilarious image!), there are two basic ways to convince your DM:

• Real world: make them cookies, do their laundry, etc.
• Game world: pay enough GP, go on a quest to find a suitably qualified inventor (a Dragonlance Gnome Tinkerer springs to mind), do it yourself (using Smith's tools), etc.

In any case your DM can put serious limitations on the fake armor, like it's difficult to move in, or one hit and it falls off. Your DM might also refuse outright if it doesn't fit the tone of your campaign.

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.