After amending, there is no change in federal refund but due $1 to state, following questions remain

After amending, there is no change in federal refund but due $1 to state,

1: should I also e-file the amend document to federal? It seems that Turbo tax will e-file it too although there is no difference in federal refund (and I don't due any money)?

 

2: for state, I need to mail the amend document, and the $1 check. Does the amend need to be processed and accepted by April 15th? Or I only need to mail it to the state by April 15th.

 

Thanks for every help!