1. There is no such requirement that I am aware of at this point today (June 2017) that you have to notify any US government agency or Indian government just because you received your Green Card. However, what you should do is you should inform your employer that you have got your Green Card so they can update your Form I-9 which is an internal matter within the files of the employer. They don't have to update any government agencies.
2. Now, traveling has certain predicates or certain dependencies that should be calculated and that should be known. First of all, people have this idea that if I leave the USA for less than 6 months I won't lose my Green Card. Well, its little bit more complicated than that. Technically, if you leave the United States after you got the Green Card, you leave the United States with the intention to never come back even if you are gone for one day and the second day you change your mind and come back. If it could be proven that you left with the intention to never come back your Green Card is gone. So, the first thing very important - your intention. If you have to travel outside the USA where your intention is never to come back, your Green Card technically can be lifted.
3. No problem at all, especially when your home is in the USA, you live here, you work here, your children go to school here taking vacations for a month, two months even three -four months is not the problem. So, some of these factors are important. Sometimes what happens is, let’s say you are in the USA and decide to go to India and take the job over there for two or three months and that goes up to seven or eight months you could be in trouble because the government can say that it looks like you quit your job here. The job is important, where your children are going to school is important, your permanent home is important. These are not issues for most of the people who live in the USA but I am just pointing out these so when a situation arrives I want you to consult a lawyer before you make a plan.
An Intention is a state of mind, the state of mind cannot be directly looked at it can only be inferred from circumstances. So circumstantial evidence is important to prove your intention. For example, if you leave the USA with a one-way ticket and you don't come back for one year, chances are you lost your Green Card because if you are outside the USA for more than one year you will automatically lose your Green Card.
There is something called Reentry permit which allows you to go up to two years and the reentry permit is a definitive announcement by the US Green Card holders that I do not intend to give up my Green Card. It is not the full proof for saving your Green Card but it is quite effective, if you need to go away for two years you can file for a reentry permit.
For reentry permit, you have to be in the United States to file for it. After you file it within a few weeks you will be scheduled for Biometrics. You have two choices whether you file and leave the USA, you have to be physically present when USCIS receives your application and then you can leave and come back for the Biometrics or you can stay here, do your Biometrics and then you go. The choices for receiving your reentry permit when it is approved are either to your lawyer or to the address in the United States or to the Consulate of your home country. If you go to live in Mumbai, you could pick up your reentry permit in Mumbai Consulate.
If you face a very bad emergency when you are in India, you can always apply for reinstatement of Green card which is called Returning Resident Permit or SB-1 Visa, that is done through the US Consulate. You go to the Consulate, explain to them in writing through the form. Everything has a Form, that shows that your stay was unintentional, they can take time and reinstate your Green Card for you. There is another Form you need to fill in, that's called Form N-470.
4. N-470 is a highly misunderstood Form. N-470 is helpful in naturalization only to the extent that normally, if you are gone for one year you have to start naturalization five years all over again. Normally, you can apply for Naturalization five years after you get your Green Card. If you need to stay away for one year you have to start your five years all over again. By filing N-470 you prevent that re-setting your five-year clock and there are very specific rule that covers N-470 including that there must be one year during which after getting the Green Card you must not have traveled outside the USA even for one day, this is one odd little rule but keep all these things in mind. So, N-470 is usually filed in conjunction with reentry permit application.
5. No, of course not.