Thank you for your question. I look forward to working with you are seeking for educational purposes only.
I am afraid that an employer has a right to expect an employee to be on time for work. 2-3 minutes late is not on time and an employer has a legal right to discipline an employee for even such slight tardiness, especially if it is consistent. It is legally not the employer's duty to make accommodations for an employee to get to work, that is solely the employee's responsibility I am afraid.
Legally, if you were injured from lifting at work, you are entitled to file a workers compensation
claim, even now that they fired you, and you are entitled to treatment under workers compensation. The employer cannot refuse to file a report of injury, although they can later try to dispute
your claim based on you not filing an actual report of the injury when it occurred. Unless you told your manager you wanted to file a workers compensation claim for your shoulder, he was not bound by law to do anything, so it would be hard to prove this termination
was retaliatory for filing a workers compensation claim. Had you tried to file a claim and the manager refused and fired you, then you could have had a potential claim for retaliation
based on you filing the claim.
As far as termination, while you feel you were not treated properly, legally anyone with authority to do so at the employer can give an employee notice of termination and it did not have to come from your manager personally.
Under CA law, unless you had a contract to the contrary, your employment is at will
which means the employer can terminate an employee for no reason at all or any reason not based solely on the employee's age/race/sex/disability/national origin and the employee's sole recourse is filing for unemployment
and that is it.