The Animated Shield says that it
protects you as if you were wielding it.
This strongly implies that you aren't wielding it.
The Monk's Unarmored Defense feature works
while you are wearing no armor and not wielding a shield.
So the use of an Animated Shield shouldn't interfere.
As for proficiency, the only requirement for using an Animated Shield is that you're attuned to and holding one, so this shouldn't be an issue. There is an argument against this, though:
The Animated Shield protects you as if you were wielding it.
Luckily, wielding a shield with which you're not proficient doesn't affect how it protects you; all it does it give you a bunch of penalties to other things.
Since the Animated Shield "protects you as if you were wielding it", and doesn't "give you penalties as if you were wielding it", this isn't an issue.
The “x will happen as if y were true” construction is used several times in the rules, to lay out specific cases for a rule to come into effect. In all these cases, a particular thing can be done as if a prerequisite were true - but that prerequisite remains false.
Anything not specifically covered in the “as if” statement remains covered by the general rules.
Jeremy Crawford tweet calls out the “as if” statement in his clarification that it is possible to use feats/abilities that tie into the "as if" statement:
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.
Spell Dodge and Bash: A Split Decision
Protect is not a well-defined term in the rules, so I would rely on its everyday definition: keeping you from getting hurt.
If you were wielding the shield, you could protect yourself from spells (since you have that feat). So the animated shield can enable Spell Dodge.
Shoving or bashing is not, strictly speaking, protecting yourself (your vice principal never accepted that, and neither to do I). So the shove/prone bonus action cannot be used through animated shield.
Further Reading: Examples of “as if” in the rules
I get my understanding of the “as if” clause from other cases in the rules, including the following examples.
For darkvision, the two cases are carefully laid out.
The monster can see in dim light within the radius as if it were bright light, and in darkness as if it were dim light.
D&D Basic Rules, p. 4
The rules don't say, “for a monster with dark vision, darkness is dim light, and dim light is bright light” — because that would equate darkness with bright light.
An elf can see in darkness as if it were dim light, but it doesn’t make the area dim light. Creatures (including those with darkvision) that have powers which works in darkness can use them, whether or not someone can see them. Darkness doesn’t “become” dim light for an elf, only the specific case of vision is affected.
For multi class spell users, the as if clause covers two cases that are delineated: spells known and spells prepared. The other facets of the spell casting rules (spell slots, etc.) are described elsewhere in the section.
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
PH, page 164
Although shields of any kind are not considered body armor for the purpose of AC calculation changes (as per your previous question), an animated shield still counts as a wielded piece of equipment with whatever downsides that includes, with the exception that it keeps your hands free for spellcasting or whatever else.
Shield proficiency is listed under Armor Proficiencies and as such has to be taken into account in this case. RAW heavily suggests that you still take the full penalties of wielding a shield, including not being proficient with it, while using Animated Shield.
See this question for a more detailed explanation.
EDIT: As Mindwin pointed out, by a narrative standpoint, not having the knowledge of how the shield you're using will react to attacks to attempt to deflect them (not being proficient in its use, animated or not) would make it extremely distracting and hard to move around with, specially if you need to perform complicated maneuvers and mantaining focus during spellcasting or in a usually frantic combat situation.