field theory - Is it enough to assume $F_{munu}to 0$ at infinity but not $A_mu$ to derive the equation of motion?

Suppose the the Lagrangian $\mathscr{L}$ of the free electromagnetic field is augmented with the term $$F_{\mu\nu}\tilde{F}^{\mu\nu}=\partial_{\mu}(\epsilon^{\nu\nu\lambda\rho}A_\nu F_{\lambda\rho}).$$ Since this term is a total divergence it can be converted to a surface integral over $A_\nu F_{\lambda\rho}$.

Now, if $F_{\mu\nu}$ is assumed to vanish at the boundary (at infinity) it is sufficient to argue that such a term drops out in the equation of motion (EOM) this abelian theory even if $A_\mu$ is nonzero.

Questions

1. 
   

   Is $F_{\mu\nu}\to 0$ at infinity sufficient to derive the EOM in absence of the $F\tilde{F}$ term? Did anywhere in the derivation of the EOM we need to use that $A_\mu$ itself have to vanish at infinity/boundary?

   
   


2. 
   

   As a related question, physically, why do we need $F_{\mu\nu}$ to vanish at infinity?

   
   

Next, if we augment the Lagrangian of a nonabelian Yang-Mills (YM) theory with a $F\tilde{F}$ term, unlike the previous case, it does affect the EOM unless $A^a_\mu$ itself are assumed to vanish at infinity.

3. Do we assume $A_\mu^a=0$ at infinity only for a nonabelian YM theory or do we always have to assume $A_\mu=0$ irrespective of whether the theory is abelian or nonabelian?