B-1 Visa

I have known many immediate relatives of the U.S citizens and they have never had a problem. However, theoretically the problem exists. It would be right to do an immigration pre-clearance somewhere close to home.

Regarding your query on medical emergencies, you could tell them that your parents took the medical treatment. As long as there was no government funding I do not see any problem.
 

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I would strongly recommend against it. I think you would do a lot better trying to get an H-1B Visa stamping in a third country. People are going to places like Singapore, Hong Kong and Maldives. Look around India and you might be able to make an appointment. 
 

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Note: Unless the context shows otherwise, all answers here were provided by Rajiv and were compiled and reported by our editorial team from comments, blog and community calls on immigration.com. Where transcribed from audio/video, a verbatim transcript is provided. Therefore, it may not conform to the written grammatical or syntactical form.

1. Your obligation is complete when the government physically receives your papers with the proper filing fees. After that, if they delay in creating the receipt it's really not your problem. As long as you can show that the papers were sent and received by the government. 
2. The answer is No. Let's say that they did leave, then they’re I-130 would still go on but their I-485 would be abandoned and you would have to convert the case to consular processing. 
3. They are automatically eligible to stay because once their I-485 is timely filed, they are considered to be an authorized period of stay. They cannot be illegal as long as the I-485 is pending.  
 

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Note: Unless the context shows otherwise, all answers here were provided by Rajiv and were compiled and reported by our editorial team from comments, blog and community calls on immigration.com. Where transcribed from audio/video, a verbatim transcript is provided. Therefore, it may not conform to the written grammatical or syntactical form.

The Customs and Border Protection (CBP) people do not hassle the parents or spouses of the U.S citizens too much as long as they are not entering to adjust to green card status within the United States. But if a CBP officer is either ill-informed or particularly unnecessarily harsh they could stop you and ask you to go back (which is unlikely). 

A practical solution I would recommend to people is to do an immigration pre-clearance in Abu Dhabi (if you are traveling from India) that way if they are denied entry they just return from Abu Dhabi they don't have to come all the way to the United States.

Note: Where transcribed from audio/video, this is a verbatim transcript of the referenced audio/video media delivered as oral communication, and, therefore, may not conform to written grammatical or syntactical form.

In this case it should be fine, but remember to go through the Section 222 (g;) of the Immigration and Nationality Act. Another thing you need to remember is in order for you to get a B-2 visa stamp or a B-1 visa stamp at the consulate you do not need any kind of approval from the USCIS.

Note: Where transcribed from audio/video, this is a verbatim transcript of the referenced audio/video media delivered as oral communication, and, therefore, may not conform to written grammatical or syntactical form.