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