Termination based on misbehaviour

Hi, My employer did not provide me with any appointment letter neither any offer letter at the time of joining and till date. I have worked with the employer for 5 weeks without hesitation just with a trust of word from mouth that I would receive it soon. I have enquired and stressed on the fact to provide me an appointment letter at least several times during this period, to which the replies were irrelevant and said me that this is their company policy to provide it later. I forgot to mention that I was appointed as Process Trainer for this startup. I received an email mentioning about my date of joining, else nothing nor even a single document that states their policies, terms and regulations. I have ad argument twice in regards to the same. Latest being the friday of 19th May, I had a verbal argument of not providing an icard nor a proof of employment with the employer and working at night shifts, to which I have been couple of times stopped over by the police patrolling at night. They issued me a termination letter stating for my misbehaviour. I find this absurd since stating my demands and rights for appointment letter, they have out thrown me from the company. Additionally, I still owe my pay from the 1 - 22 of May (the date this letter states the termination from). Also I have not signed their termination, stating a fact that this letter stands nullified since I have not received any offer nor appointment letter but they have disagreed stating that the email itself is an appointment letter. In regards to the misbehaviour, I was never communicated verbally nor in written by them. I am in a dilemma now! Should I send them a legal notice through a lawyer ? Question 2: If I did what is the adverse affect on my career ? I am young just 26 with wife. Question 2: I am not that financially strong to pay my lawyer. So are there ways the lawyer can understand my circumstance and the mental harassment I am going through right now in addition to my financial condition with loan and a married life with expenses ! Please advice.