1. Do I need a new LCA as my current LCA is expiring on 1 May 2017?
Yes, your E3 is not valid for work without a Valid LCA
2. Which expiry date should I need to follow to stay in US legally? VISA STAMP/LCA/i 94? I remember advocate told me to follow the i94. But does anyone had a similar issue?
Actually I-94. But to work, you need corresponding LCA and proper VISA OR APPROVAL FROM USCIS. You may have to file an I-129 Amendment request with USCIS as the I-94 could be an error for 2 years without a valid LCS at the time of entry. The burden lies on you and your employer to correct any inconsistencies.
3. Am I legally allowed to work in US when LCA is expired but still my i94 is valid up to Mar 2019? (I am not travelling out of country after May 1 2017 as reentry is not possible because my visa label is upto May 1)
NO, you can not work without the LCA. LCA takes only 7 days to get certified.
Forms Preparation fee for H1B/E3 I-129 (Extension/Transfer, Amendment) is $295.00 (includes Public Access file), H1B/I-140 RFE write-up Starts at $195.00, H4 EAD $95.00 and PERM/I-140 starts at $500.00, check us out at www.vesaservices.com or email at email@example.com