you can be sleeping partner in partnership firm 

however before joining firm as sleeping partner obtain prior consent of authorities 

Yes, a government employee can invest in a limited liability partnership/partnership but being a partner it is not mandatory to take part in the daily affairs of the firm.

He can only be a sleeping partner and cant be a whole time.

1. Govt. Employees are barred from any conducting any "financial ventures" of any kind, in anyway whatsoever & which-so-over....

2. For violation of above, punishment upto dismissal is applicable.

No, you are in full time employment and as per service rules you may not start any business as partner of register a business.

There are restrictions for a government employee to do business simultaneously in any form or mode.

Sleeping partner  also  means that you are involved in the business.

Being govt employee you can't as it will be dual employment

No, A government employee is not allowed to run a private business, neither is he allowed to work anywhere else as a part-time or full-time employee. This is against government rule and hence person who is found can be charged for breaking the law.

See in terms of your employment rules you can obtain the requisite permission from the appointing authority and can start the firm as sleeping partner.

 No you cannot do any form of business whether it is partnership or proprietor ship firm. 

1) you are govt servant you cannot run coaching center 

2) you need licence under shops  and establishment act 

3) you need to open bank account in form name  fulfill KYC formalities 

4) registration under GST if turnover is over Rs 20 lakhs

Coaching of what faculty ? Non professional course, no registration required.

If you want to open a small scale coaching there is no need of license but for opening coaching on a large scale, that is, it intends to be the size of an institution then, you needs licensing. You need to get a trade license and pay tax for the revenue generated through the centres. If the financial gain of the centre exceeds 9 lakhs per annum, registration of the business becomes mandatory and service tax becomes payable within 30 days.

There is no specific law or provisions of law for establishing online coaching center.

All businesses are required to be registered under the Shops and Establishments Act with the inspectorate of that area. This must be done within a month's time of setting up the business- subsequent to which a statement is to be issued to inspector.

But, if your coaching centre intends to be the size of an institution, you would need to get a trade license and pay tax for the revenue generated through the centre. If the income exceeds 9 lakhs per annum, registration of the business becomes compulsory and service tax becomes payable within 30 days.

See legally you cannot do such business wherein you participate full time or part time. For coaching institute as such no compulsory registration is required. 

Nothing just a place you need to get shop and establishment license. If no fixed place then no need