Thank you for your patience. I cannot rush my answers since I try to be comprehensive with each question and the follow-ups asked. That is how I have attained such an excellent rating with customers on the site.
My concerns are following
i)When my current new company starts my GC processing would the part-time H1B petition be a reason of concern.
No, one has nothing to do with the other. The fact that an old petitioner finds a need for your services has no impact on a new petitioner filing a new PERM and I-140 for you. Both of these are PROSPECTIVE in nature for a future time when the priority date is current. Employment circumstances change because of the long delays and working part-time for an old employer is not negative in any way.
ii)When I go out of US and go to consulate for Visa stamping would there any issues
Again, no, there is no negative inference for having a concurrent petition for a prior H1B employer. It is all lawful and legal as long as your FT principal H1B petition is a real job offer and you are working it and have a payroll history to show.
iii)Would I be required to let my current employer know that I am working part time.Am I legally allowed to do that
You are legally allowed to have a concurrent petition. That is totally why the law provides for concurrent petitions. Yes, you are allowed to have a concurrent H1B.
iv)Infact my new company's H1B transfer has not been approved yet and its still in Initial Review state.Can I start my concuurent H1-b processing now.
NO, you could go back full-time to Co A and work under their approved full-time petition but going to work for them part-time whether it is concurrent or under their prior petition would require an amendment or concurrent petition approval because the petition for Co A is for full-time. You cannot work under that petition part-time. That is a material change in the terms of the petition and requires either an approved amendment or concurrent petition.