EdDSA Verification vs. Cofactorless Verification
1. It means that someone possessing the private key $s$ can create signatures which pass verification but not cofactorless verification with respect to the public key $A$.

3. For example, if $C\in E$ is a point of even order (e.g. $2^c$), and the signer replaces $R:=rB$ in step 2 of the signing procedure by $R:=rB+C$, then
$$
  2^cR+2^cH(\underline R,\underline A,M)A
  \;=\; 2^cSB+2^cC
  \;=\; 2^cSB
$$
but
$$
  R+H(\underline R,\underline A,M)A
  \;=\; SB+C
  \;\neq\; SB
  \text.
$$
The private key is necessary to create such a signature because anyone modifying $R$ (or one of the other parameters) after the signature has been finalized (e.g., someone who does not know the private key) is very likely to be caught by $H(\underline R,\underline A,M)$ contributing to the verification formula. (Note this contradicts Cryptostasis' comment above.)

2. Due to the Pohlig-Hellman algorithm, the security of the whole system is dependent on the large prime-order subgroup $E[\ell]$ of $E$, while small prime factors of the group order do not reasonably contribute to the hardness of computing discrete logarithms.
Group-theoretically, multiplication by $2^c$ realizes a projection to the subgroup of $E$ of size $\ell$, which is a direct factor: From the structure theorem of finite abelian groups,
$$ E\;\cong\;E[\ell]\times G\text, $$
where $G$ is an abelian group of order $2^c$, hence multiplication by $2^c$ maps an element to $E[\ell]\times\{e\}\leq E[\ell]\times G$. (And since $\ell$ is coprime to $2$, multiplication by $2^c$ is an automorphism on $E[\ell]\times\{e\}$.)
Thus multiplication by $2^c$ throws away a little bit of information from the factor group $G$, but that wasn't useful anyway.

In practice, it is not going to matter which variant you use. As explained above, cofactorless verification is a little stricter, hence you might run into interoperability problems with non-conforming signers (but I see no reason why anyone would implement it in a way that makes this happen). On the other hand, verification (with multiplication by $2^c$) will take a few cycles more.