I am working on H-1B. The job will end in a few weeks. What are the implications of the "60 day grace period" rule that has been created recently?

1. For a resident who is completing residency on Jun 30 on cap-exempt H-1B visa and has a job offer from a hospital that is cap-exempt because it is non-profit and has affiliation with a medical school for training, can the employer be requested to first apply for a cap-subject H-1B visa on April 1? <br>
2. If the lottery does not come through, can hospital apply subsequently for cap-exempt? Are there any risks with this approach?<br>

3. Is a start date of July 1 possible with cap-subject?

My EB2 priority date is June 2009. I filed for my I 485 in 2012 and since 2013 I have EAD and have been maintaining H1b status.
In 2014 I changed my job from the original employer who started my GC process (with whom I have I 140).
I sent AC21 when in 2016 I got the standard RFE to produce medical records and employment proof (since then the priority date had retrogressed again), and also have H-1B with my new employer. I have not started the new GC file with a new employer.
Should I file supplement J now, proactively, or should I wait for another RFE from USCIS when my priority date becomes current in the future, to file for supplement J?

My I-140 approved (India/EB2) from Employer A in June 2015, I joined Employer B in Oct 2016. Employer A sent revocation of my I-140, on USCIS website today is showing below status: "On December 7, 2016, we received your correspondence for Form I-140 ..." Will I still be eligible for 3 years of multiple H-1 extension with my approved I-140 from employer A without filing new I-140?

My elder son born in India and now he is 10 years old. We came to the US in 2009. My visa status is H-1B and GC is in progress. My son is a bleeding disorder patient(like hemophilia). The medicine available only in the US. Does my son's medical problem make any issue while applying 485/EAD or at the time of GC issue?

I am working for same company for more than 9 years out of which 2007 to Aug 2014 in the US and since Aug 2014 in India Development Center for the same company. I have visited the US frequently (4 times) between Aug 2014 to Sep 2015 and in India since then (through Advance Parole). I received my GC on 6/26/2016 and have got the physical cards delivered to me in India by a friend traveling from the USA. My father was sick for last one year and passed away on 15th Nov 2016. I have EAD card with validity till 12/10/17 (Not Valid for reentry to the US). My Advance Parole has expired on Dec 29, 2016. Also, 6 months timeframe for GC entry to the US has expired on 12/26/2016. <br>
I want to work and live in the US but for immediate time (next 1 to 2 years) I need to spend more time in India to console and support my grieving mother. My current company does not have any immediate work for me in the US. I am looking for options outside my company to come back to the USA.<br>
1. Can I directly join some other company in the USA using the same GC my company has filed for me? If yes, are there any legal formalities/paperwork involved to do that?<br>
2. I have already filed for Advance Parole twice between 2014 and 2016. Once I am in the USA, can I file for a Re-entry permit to get a leave for another 2 years outside the USA to support my mother while she is going through recent grief? <br>
3. How long does the process for Reentry permit application take? Do I need to stay in the US till the time application is approved or can I leave for India once application is filed and bio-metric is done?<br>
4. On a side note - I have no earnings in the US for the year 2016 but I must file the taxes as a GC holder (and also to maintain my ties with the US). Do you have references for any good CA who can help me with this?