Running firm without registering it

It is mandatory to register a company / firm. Nobody will purchase your product if there is registration. And you will also be penalised if you are carrying out business without registration.

And also, you cannot sue or be sued as there is no existence of the same in the eyes of law.

IF THE INTENDED FIRM IS PROPRIETORSHIP, THERE IS NO REQUIREMENT OF ANY REGISTERATION EXCEPT OBTAINING A VALID TRADE LICENSE FROM THE MUNICIPAL AUTHORITIES. THE ANSWER TO THE SECOND PART IS AFFIRMATIVE.

Only formation of a Company needs Registration. Even a partnership firm can legally run its business without Registering it.

The founder can accept the payment via cheque in his favour but if it is basically a partnership firm then he is liable to divide the payment amongst its partners.

Is there a way to check the registration of a company in MCA (Ministry of Corporate Affairs) website?

https://www.mca.gov.in/DCAPortalWeb/dca/MyMCALogin.do?method=setDefaultProperty&mode=16

You can check by clicking on the above link.

If you wish to carryout your business as a proprietory concern or as a partnership firm the option is with you whether to go for registration or not.  But if you need to operate as a company registration is a must.  There is no bar for the owner (founder in your words) to accept receipts or make payments through cheques.  Than the unregistered, the registered firms or business concerns enjoy public trust and make good business comparitively.   

Thanks dear experts.

Any business concern requires registration under different legislations. So far as form of constitution of a business concern is concerned, only a company (whether be a private limited or a public company company) is required to be compulsorily registered. Neither a proprietorship concern nor a partnership firms requires compulsory registration for running a business. However, a business concern may require registration under other legislations such as under labour laws, municipal laws or other laws regulating other aspects.

to do any business as sole proprietor, as ENTITY registration is not compulsory but under sales or excise/ service tax laws it's required to get registered with a specific banner name proprietor can accept the payment via cheque in his favour