Can a working mom get fired from her job for her child being very ill, even if she has a dr note?

Unfortunately, you do not qualify for rights under the Family Emergency Medical Leave Act. The place doesnt have enough employees to make it comply with the law, and if it did, you need to be working there for a year to qualify. So the answer is yes, you can be fired for missing time at work.

I would consult a lawyer who specializes in employment law in your State. I would doubt it is so cut and dried.

Yes, the size of your Company does not qualify for protection under the FMLA and you are in an employment at will state.