I work in a regional level 3 TC, with many outlying rual areas and two major highways going through our town, brining in lots of out-of-towners...here's how things work in our area for unresponsive/confused patients:
Our 1st responders generally don't have time to check for ICE, but if LE is on scene, someone has generally at least looked for an ID for the patient by the time they get to us.
Once in our TC, someone, be it a nurse, chaplain or SW, will go through whatever belongings arrive with the patient. We first look for ID, so that we can see if they've ever been a patient at our facility before. If they have, bingo, we may have all the med hx we need on our computer files.
Second, we look for a next-of-kin contact. If we found med hx on our computer, we may not contact next-of-kin for a while unless the patient is critical or deceased. If we have no hx, an outside family/friend contact becomes a more immediate need. We do try to contact someone, but it may be put on the back-burner until the patient is more stable. We also have to take into account the news we are delivering to a stranger however many miles away with a long drive ahead of them to reach us.
We have used cell phones. But you have to remember that this is difficult if the person was in a wreck. I've had to clean the blood and mud off a phone before to see if it even works...often it doesn't, or us old nurses don't know how to use it. ICE is a great idea (I have it on my phone), but if you are traveling outside your home area or have people/animals depending on you on a daily basis, I would recommend keeping an emergency contact list updated in your wallet, too. Cell phones may land in the ditch, but your pants will come to the ER with you.
As for service animals, I've never had one come to our ER, but we have had many patients come in who were traveling with pets when an emergency/accident happened. In our area, LE generally coordinates care of the animals. That may be with the local SPCA, or an officer may take an animal home. I would think they would check a service animals gear for info, but couldn't be sure.