In today's world the EMT crew would have cleaned-up Zimmerman prior to being allowed in the squad car. Why? To qualify and document the injuries, and because of the possibile exposure of AIDs and other transmittable diseases be transported in the police vehicle.
Many domestic violence cases do not involve EMT as a matter of fact, even if the one going to jail is injured. Supressing the evidence and boosting the case ASAP is part of the MO of cities.
What you are mentioning about EMTs always there is for COP shows on TV.
I was thinking they probably took his clothes at the scene for evidence..