Role and function of (Indian)Embassy of a country

Embassy is the main link between the two countries for all diplomatic connections. It is the sole representative of a country in the host country. The point wise answers to the query are as follows -

1. An Embassy can be explained or defined as the diplomatic representation of a country's (say India) government in another country (the host country).
2. The embassy establishment is headed by the Ambassador who is working under the Ministry of External Affairs and follows the guidelines in its administration as stipulated under the Indian Foreign Services.
3. The Embassy is a vital link between its home country and the host country and has many roles. The main roles are to communicate between the two, inform the home government regarding social, political, economic and military developments in the host country, it coordinates and manages international treaties and state visits and promotes home country culture in the host country.
4. Any Indian citizen or person of Indian origin (PIO) can approach Embassy in connection with any problem experienced by him in the host country.
5. The persons working in the Embassy are under the administrative control of the Ambassador and their powers are derived from the Indian Foreign Service Rules and as per the the provisions of Delegated Financial Powers of Government of India's Representatives deputed abroad. Some other additional powers are exercised as derived from Passport Act of India and Consular Manual of India.
6. If an eligible person does not get help from the Embassy then he has to write or appeal to Ministry of External Affairs and send a copy of same to Embassy also. This happens rarely as most of the things are sorted out at Embassy level.
7. Embassy will provide guidance to the people of its home country in many spheres. For example if we lose a passport in any country then our Embassy or Consulate there will provide a replacement passport enabling us to come back to our country.
8. It varies from the country to country and their inter relations. For example India is having about 89 Embassies and 108 Consulates spread all over the world.

My answers to your points are as follows
1. An embassy is a government body of a country situated in another country which acts a link between the two countries. This is the official agency in the host country to take care of all matters between the two countries.
2. The Head of the embassy will be Ambassador. He will be the in charge of the embassy and he will report to the ministry of external affairs of his country. He will be an IFS officer and all rules and regulations applicable to IFS in their country will be applicable to him also.
3. This office is the place where all the dealings between the country and the host country will be dealt with under the guidance of the External Affairs Ministry of its country. For example, an Ambassador of India in the US will be working under the Ministry of External Affairs of India. He has to take care of all the issue between the two countries and he should take care of the problems of the citizen of his country of origin staying in the host country
4.. Any citizen of the country can approach the embassy regarding any issues he has with the host country. For example, a tourist goes to the host country and he lost his passport. Immediately he has to contact the embassy.
5. The employees in the embassy are from the country to which that embassy belongs. For them, the boss is the Ambassador and they will be governed by all the rules and regulations applicable to IFS officers. They will also have some additional responsibilities and powers as decided by the Ministry of External Affairs from time to time.
6. You have to contact the Ambassador. If there is no response you have to write to the External Affairs Ministry of your country and give them all the details and never forget to mark a copy to the External Affairs Ministry and Minister. These days some good Ministers are responding even to your tweets. So you can also tweet them the problem.
7. The embassy is the representative of your government in the host country. So they have to respond to the requirements of the people of their country in the host country. Any problems related to passport or Visa or such other issues one can approach them
8. No. It depends on the relations between the two countries. The policy will vary from country to country.

1. What is an embassy of a country(India)?
In simple terms an embassy is a place in a country with officials of another country. For instance, the India embassy in England will have Indian official geographically present in a designated place (India House, Aldwych, London WC2B 4NA, United Kingdom) in England.

2. How is the head of an embassy of a country(India)?
The top most official is often referred to as Ambassador or High Commissioner. For instance, the Indian High commissioner to England is Mrs Ruchi Ghanashyam. Usually, the officials are diplomats or with an Indian foreign service background.

3. What is the role and function of an embassy (India)?
The main role is protect the core values of interest of the country and the people that it represents. For instance the Indian embassy in England will help Indian citizens to get consular services, guidance of rights and legal advice. For instance, there was a recent burning of our National flag in Jan 2019 at London. The Indian mission (embassy) complained and lodged a protest.


4. Who, when and for what can a person or a nationality approach their embassy(India)?
Any Indian citizen, holder of Indian passport, Holder of OCI or PIO cards (overseas citizens of India, People of Indian origin), new born children of Indian parents can approach them anytime with prior appointment. In case of dire emergencies when life is at threat, an Indian citizen can seek emergency assistance at anytime.

5. What is the privileges of an embassy and the person working in it.(India)?
The basic privilege is that of a certain degree of immunity or exception from the local laws governing the country in which the Embassy is located. For example, to the common man, the Indian embassy in England, is the extension of India was far as basic rights are concerned. The UK authorities cannot enter the embassy without follow the existing protocols. Any attack ( verbal or physical) on an embassy amounts to attack on the country that the embassy represents. Such events are taken seriously. For certain officials and packages full diplomatic immunity can be provided under specific circumstances. This is often misused in the past during the Cold war and by certain countries.

6. What to do, if we don't get help from our embassy(India)?
If a person is unable to get help, he or she needs to contact the ministry of external affairs. Most embassies have a system of complaints and grievance readdressal. For instance, if you not happy about the help from Indian embassy in London, you can complain at feedback@hcilondon.in.

7. What all service do an embassy provide for its people or nationality.(India)?
Most of the services provided are visa services, help in extending passports, applying for new passports, grant of emergency travel documents, assistance/guidance in case of detention or arrest. It also offers services to journalists, film/TV crews etc. From time to time, it issues guidance and travel alerts to Indian citizens and Indian origin people regarding travel to that particular country in which the embassy in located. For instance, this is the travel guidance given by the Indian embassy in London (https://www.hcilondon.gov.in/pages?id=eyJpdiI6Im1FXC9NS1FXYUp1Z1gyZGNWak82RW5BPT0iLCJ2YWx1ZSI6IkpiazhsaXl4VERGZEtzWThQZnNjbGc9PSIsIm1hYyI6ImYwZDYwNTc3MTIyNzg4NTJhNTUxNjU2YzRmMWNiZDRmNGI1YTZhYjE0NzQ5OWJhYTBlNmY4ZGY4NDE3ZDQ5ZGIifQ==&subid=eyJpdiI6IkJwdEwycVdoZzVvKzJ5bVwvR0R0clV3PT0iLCJ2YWx1ZSI6Imk4c1RabURjbkpTYXBSZkNTSlBpTWc9PSIsIm1hYyI6IjJlZDA4YmFlYzBmNTI5YTliZjlhOTU3YzFlYWI0OGI4YjM4ZDRmMzRjYjYzNzM1Yzg0M2QxOWE2ZjRkYzQ3NmUifQ==)

8. Do all country have their embassy throughout the world?
Most countries have but there are exceptions. For instance, USA does not have embassy in Bhutan, North Korea, Iran. Some countries are so small and or so poor that they cannot afford to have embassies all around the world. Some countries, instead of a full embassy, have a simple office that accepts documents for it's citizens.

Additional reading
This is a brief information about the Indian embassy, the staff, pay etc in Israel (https://www.indembassyisrael.gov.in/pdf/RTI_info_and_Annexure_II_III.pdf).
You can also refer these books.
The Modern Diplomatic Mission by David M. Malone
The Oxford Handbook of Modern Diplomacy
Edited by Andrew F. Cooper, Jorge Heine, and Ramesh Thakur