Yes.
Although you can e-file your federal and state taxes together, if you choose to e-file your federal first, you'll have to wait until that return has been accepted before you can e-file your state.
If your federal hasn't been accepted, or you printed and mailed it, you won't be able to e-file your state but you can print and mail a paper return.
Married filers: If you e-filed your federal using the joint married status, but you want to e-file your state returns separately, only one of you will be able to e-file. The other must paper-file their state return.
Yes--you can e-file federal and mail the state instead of e-filing the state. But...why do you not want to e-file the state return? If you are using a paid version of the software your fee will be the same whether you e-file it or mail it in an envelope.
The reason for mailing state is that some states also require documentation if you want to claim income that is exempt from state taxes. A good example of this is interest on government bonds that pay through an ETF. It would be nice e-filing would allow for attachment of supporting documents, but it does not.
Doing a return for my kid who has a job. Cost to e-file State return is $25 in my state. Their refund form the state is $49. I don't want to consume over half the refund with an e-file fee when at 73 cent stamp will do the trick.