The people working for the hangover clinics are basically practicing medicine without a license, which is against the law.
No, not necessarily.
I don't know the first thing about Nevada, but many states allow what is called "delegated medical practice", the specifics of which vary by state of course, but which basically means that a physician can train and authorize just about anyone to perform just about any act, as long as the physician supervises and accepts complete responsibility for the individual's training and actions.
Based on my 0.76 second Google search, it looks like Nevada is one of these states. If that is the case, then paramedics doing this may not be practicing under authority of the state EMS agency or an EMS medical director, and still be perfectly legal. In essence, they aren't working as paramedics as all; they are just laypeople that the doctor running the drunk tank hired, trained to administer whatever they're administering, and supervises.
This was the very first result I got when I searched "Nevada delegated medical practice":
"After several abortive attempts by the Board to promulgate medical assisting regulations, the Nevada Legislature took matters into its own hands and passed Senate Bill 294 on the final day of the 2011 session. The bill confirms the authority of medical assistants to possess and administer “dangerous drugs” (i.e., any medication requiring a prescription, other than controlled substances) “at the direction of the prescribing physician and under the supervision of a physician or physician assistant.”5 SB 294 authorized the Board of Medical Examiners and the State Board of Osteopathic Medicine to develop regulations further addressing the delegation of administration of dangerous drugs to MAs (as of this writing, neither board had initiated such rulemaking proceedings). The bill also cre- ates a statutory definition of “Medical assistant”: an unli- censed person who performs clinical tasks under the supervision of a physician, an osteopathic physician, or a physician assistant; and does not include a person who performs only administrative, clerical, executive or other nonclinical tasks."
