**** My new employer recently filed my H1B, now some how I am not willing to join him, Can I ask him to withdraw my H1B****

Yes, if the H-1B petition is approved your status will be changed to that status and your L-1 won't be valid anymore. You should ask the new company to withdraw your H-1B petition. While employers must pay the costs associated with the H-1B visa process, Dept. of Labor regulations allow for these costs to be refunded if you leave them. It's up to the new employer if they will want the money back. If the H-1B petition is withdrawn, there's no affect on your current L-1 status.

Yes on 1 and 2.

1) If my H1B is approved, will I loose the eligibility to work on my current L1B visa ? RESPONSE: No so long as your L-1 employer does not revoke the approved L-1 Petition. 2) Can I go ahead and ask my offering (new) company to withdraw my H1b Filing, as still my H1B is not approved. If yes, will they allow me do do so ? RESPONSE: Yes, I would ask the H-1B employer to withdraw the Petition. They definitely should withdraw the Petition (it is the company's Petition) because they should not proceed with a Petition where the beneficiary will not work for it. 3) If I will ask my offering (new) company to withdraw my H1B Filing, do I need to pay any amount to them as compensation? RESPONSE: There could be a liability of you to the Company based on any contractual commitment you made or the total circumstances. It would not surprise me that the Comany sought reimbursement for its expenses in filing the Petition, including filing and legal fees. I would need to review all of the circumstances to determine whether you have any legal liability to the H-1B employer. 4) If my new employer agrees to withdraw my H1B application from USCIS, in this case what will be the impact on my current L1B. Can I still work using it. RESPONSE: Yes and the answer is yes even if the Petition is approved. Approval does not automatically void your current L-1.