a) Is this even legal? I have to log 1000+ transactions in a month, and if I mess up, my boss gets angry, I am not a bookkeeper.
No occupational licenses are required to be a bookkeeper. Unless you have a contract with your employer or a collective bargain agreement negotiated with your union that limits your responsibilities, the employer can assign you whatever duties the employer wishes.
I doubt this even abides my financial compliance/GAAP accounting laws, etc.
If the company is not publicly held and is not under a contractual obligation to do so, GAAP accounting is not required. What you are doing isn't inherently inconsistent with either GAAP accounting or with HIPPA (for health care information) either. The tax laws merely require that a company keep records adequate to comply. It is poor business practice but probably not illegal.
(b)Just curious what my legal rights are if they ever try to fire people over this?
If you are an employee at will (and in the U.S. you almost surely are), and you aren't in Montana (which has a requirement to fire only for good cause), the only right you have in connection with a firing for bad cause is that it entitles you to unemployment benefits which you are not entitled to receive if you quit or are laid off, and there are alternative severance pay arrangements when that is allowed by state law.
Also, inability to perform assigned job duties to the standard set by the employer is good cause. So, you could be fired without negative consequences for the employer if your boss gets angry. The fact that your aren't qualified for the kind of work and bad work processes your employer wants you to engage in doesn't really matter.