Hospitals give a fuck. Its like the only privacy law that is enforced and enforced well. Fines are high and people lose jobs over HIPAA violations.
Texas can tell them to violate it but doubtful they would be successful unless they are willing to foot $$$ on behalf of the hospital. Also, likely goes against hospital mission, values, etc.
Yes, they are required to treat anyone, regardless of status. EMTALA.
For your first question, we know and if not directly told, sometimes we have high suspicions. I employ social workers who directly ask what their social situation is (housed, finances, etc) and for any uninsured we ask if they would like Medicaid. More often than not they disclose (CA is a lot friendlier than TX) and we document it and find the appropriate resources for them. If they dont tell us, we dont document our suspicions (because we cannot objectively confirm) but we can def tell. They tend to speak another language only, be very guarded with their questions, etc. We often have Chinese and Hispanic individuals undocumented. We connect them to MediCal, PRUCOL, and treat them like anyone else, providing them as much support and resources as possible.
I'm not sure how forthcoming they are in Texas but im very positive these Texas hospitals also know which active and prior patients are undocumented if that information is disclosed. It is not difficult to search charts.