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,
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. GAAP accounting (which is compelled by securities laws), HIPPA and similar laws governing record keeping and databases don't govern the mechanism by which data is put into or retrieved from databases or records (other than required security measures for the data in the case of HIPPA and some banking laws) or the precise format of databases/records.
The employer has made an implicit decision to do work in a stupid and inefficient way and to pay you to do it, rather than to do it in a cheaper and more conventional way, but that is the employer's prerogative. The employer would be perfectly within the law and its rights to have you use an abacus and cuneiform ledgers in mud with a stylus if it wished, so long as the ledgers were kept in a secure environment.
The tax laws merely require that a company keep records adequate to comply. It is poor business practice but probably not illegal. Workplace.SE would probably have suggestions about how to get an employer to be less stupid.
It may even put the company at risk of liability for negligent misrepresentation to a third party causing a customer harm occurs if an error is made and then reported to a collection agency and the error prone method is found to be negligent by a judge or jury. But, it would not be illegal and would not give rise to liability, unless an error that negligently causes harm to a customer actually happens.
Also, first party debt collections (as opposed to debt collections through a third party debt collector) are not subject to the Fair Debt Collection Practices Act. So a mistake in a bill sent to its own client would not be actionable.
(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.