Is it customary to be discharged with C. diff Infection and take antibiotics at home?

My spouse was sent home from the ER with a diagnosis of CDIFF which he picked up in another hospital after a surgery. No instructions were given to us - fortunately I have a nurse friend who told me what to do. Since you mention "home is a sterile environment" you will know what actions to take. Hopefully the home has more than one bathroom so others are not using the same bathroom. The internet has a lot of information. It was a PITA trying to remember all the protocols.

I agree with AlvaDeer about the fecal transplant.
And a side note the home is not a sterile environment and the possibility of picking up other infections, bacteria in a hospital or even rehab is far greater than picking something up at home. Other family members are at greater risk and precautions must be followed.

Can she moved to a rehab?

That is true, what the MD told you. Acute care hospitals care only for those acutely ill. C-diff can go on forever, and often is very resistant to antibiotic treatment If the person is continuing to have explosive stools despite all remedies and is in danger of losing weight you may pass into the realm of acutely ill again, and I have as a nurse seen many deaths due to C-diff. If this is resistant to treatment seek out GI MD who will treat with fecal transplant. Often immediately successful.