Everyone who can be immunised safely (no egg allergies and so forth) should get immunised. People who are not immunised are a danger to others.
AFAIK, PSA, ISI and USFSA do not require it. Does anyone know if it is common for rink managers or skating program directors to require it? I've never heard a parent ask about it, but I've never been a professional coach, so perhaps some do.
I know many people who routinely get flu vaccinations. Not so many who routinely get other vaccinations.
Rinks should have a first-aid trained staff member on duty at all times. I would guess insurance would require this.
Some rinks here (in Maryland) do not, so I assume some insurance companies don't require it. It would be interesting to know if the big rink insurers (e.g., the ones used by ISI, USFSA) do.
All coaches in Canada have their first aid. It's not the same in the US?
No. Rinks are largely unregulated. There may be some location-specific laws about having electricians, firemen/women and/or medical personnel on hand during major public events such as big ice shows, that apply to any public facility, but I don't know of anyone who regulates the presence of first aid personnel the rest of the time. One rink did not used to allow personnel to offer first aid help or advice - nominally for liability reasons.