Lets say someone arrives at ER with acute renal failure. What would be the extent of ER Doctor's treatment of the patient versus an icu specialist or some other specialist. That is tell me what an ER Doctor would do in these situations versus Critical Care or Nephrologist,etc:
For most acute renal failure issue, optimization of the patient's hemodynamics is going to be the treatment of choice; appropriate resuscitation, antibiotics, vasopressors, inotropes, etc. Whatever is appropriate to optimize renal perfusion. In that regard, there is very little that is going to differ in how the patient is managed from specialty to specialty (assuming everyone is doing their job). If you look at my procedural credentials versus an Intensivist's credentials, there isn't much that is different. As far as I can tell, I'm not credentialed to perform bronchoscopy, but that is about the extent of it.
The relationship between the Emergency Physician and the patient has to end at some time. Typically that is 4-8 hours. The big thing the intensivist has is a bunch of beds that are dedicated to critical care and nursing ratios that are more ideally suited to critical care. The other thing that the intensivist has is training for the next steps. I can manage a vented patient for 24 hours in fashion fairly similar to an intensivist. After 24 hours, I start to run out of knowledge and experience for how to manage a patient that is continuing to decline or won't wean.
Nephrologist prescribe dialysis and can often perform renal biopsies. Neither EPs nor ICU docs can provide that. They are best suited for chronic renal issues that go beyond hemodynamic management. They are also good at treating issues like Lupus, which have more complex issues surround renal injury.
If you go back to your list, 1, 5, and 6 are treated by fixing hemodynamics. 3 is really treated by good glucose/blood pressure control and is an outpatient issue. ICU, ED and nephro all treat #4 the same way; call someone else. That patient needs someone to unobstruct the flow of urine, either urology for a stent or radiology for a nephrostomy tube. As above, 2 is too broad, although many nephrotoxins are treated by optimization of hemodynamics.
And a question as to when ER hands it over to some other doctor how much does he know about further management and outcome for the patient?
It strongly depends. At most hospitals it isn't all that hard to follow up on your patients. It is requirement of EM residency programs, in fact.