No. It is equal to "if not B, then not A." You're welcome for doing your logic 101 homework for you.

An example of why this is incorrrect.

If a card is the ace of spades, it is black.

A card is black if and only if it is the ace of spades.

There are other conditions under which B (a card is black) can happen, so the second statement is not true.

A conclusion that would be correct is "If a card is not black, it is not the ace of spades.". The condition is that if A is true B will also always be true, so if B is false we can be sure that A is false as well - i.e. "If not B, not A".

If Nazi, then fascist = true

Fascist, if and only Nazi = not true

If car, then vehicle = true

Vehicle if and only if car = not true

No

You've have some examples, but in case they are not clear enough:

If [you have AIDS] then [you are unwell]

[You are unwell] if and only if [you have AIDS]

The first one is not the same as the second. Why? There are plenty of ways to be unwell, without necessary developing AIDS.

The first statement only defines one possible path to B, not all of them.

No.

If A, then B

If Not B, then Not A

If it's raining then the grass is wet, but you can't tell if it's raining if the grass is wet, because of say, a hose or sprinkler.

All that you can tell is that if the grass is dry, then it is not raining, and I that's called a contrapositive.

if youre doing homework, i recommend writing out truth tables for the statements and comparing, gives you a bit more insight into the statement truth conditions

“If X is cat, then X is mammal” =?> “X is mammal if and only if X is cat”

Obviously doesn’t hold: What if X doge?

if I brake, my Car will stop.

will my Car only stop if i brake?

I just saw a video on all the logical fallacies that exist, and this was one of them but my shit-ass memory can't recall what the name of the fallacy was.

Nope. The first statement doesn’t exclude any paths to B

A => B is not the same as B <=> A

No.

B iff A is defined as "If B then A and if A then B".

If that doesn't make it clear enough for you, then try writing out the truth table for both statements.