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That is good when the supply is dual polarity. If the supply is only one polarity then use a voltage divider to set the non-inverting input at half the supply voltage.
Yes that is safe.,
Some opamps might require you to tie both inputs to one of the rails (depending on it's architecture) and leave output open (preferred to keep the current consumption low). But when no such thing is recommended by the manufacturer, buffering the ground is the safest thing to do, but make sure that your input common mode includes ground.
Many older FET-input opamps like the TL07x have a problem called Phase-Inversion. When one or both inputs have a voltage that is lower than the input common-mode voltage range then the output suddenly goes high.
I don't think it does harm when an opamp output is forced high or low by its input voltages.
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