My sister works for a large daycare chain in WA State. She needed to have her wisdom teeth pulled and was in so much pain she was in tears and shaking. On a Friday at lunch she asked if she could go to the dentist so she didn't have to go through the weeknd with the pain. They refused to let her go. Then if you call in sick 3 times - even with a doctor's note - you will be terminated. Is this lawful?

Unless a person's absence is due to a serious illness that is covered by the Family and Medical Leave Act (FMLA) and both the employer and the person meet the eligibility requirement for FMLA, a doctor's note has no force in law.

so there isn't any protection for an employee if they get sick/hospitalized unless FMLA plays a role?

That's correct!

Barring a bonafide employment contract or CBA (union) that states otherwise, a termination under the terms you describe would be considered legal.

so there isn't any protection for an employee if they get sick/hospitalized unless FMLA plays a role?
I would think if someone was hospitalized, it would be a different story, depending on why they were hospitalized, but I guess without knowing that you would be hospitalized, and filing for fmla, then I could be wrong.

FMLA (and the WA equivalent) is the only job protection in the poster's state. Hospitalization is irrelevant if the medical condition, which includes the time off for the condition (minimum of 3 days) doesn't qualify for FMLA.