When TurboTax prepares Form 1310 for your deceased MIL's return, it will be included in the Efile of the Federal return. In your PDF file, if Form 1310 looks correct, no other signature is needed. I'm not sure what you mean by 'TT is Efiling on my behalf'? The Federal return is being filed by your wife (as Personal Representative) on her deceased mother's behalf.
I would recommend having the refund direct deposited into an account that has your wife's name on it. If she is on the same account as her mother, it should be fine to direct deposit it there. If she isn't, the refund should be deposited in an account that does, or the bank may refuse the deposit.
Yes, your wife can sign as Personal Representative on the mailed Illinois return (per the Form 1310). The same advice applies to the IL Refund as Federal refund.
Sorry for your loss.
@carriskoul
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