Can contractor start the project without his and board signature on the contract,

It sounds like you have some reasonable concerns. Why should your board allow the contractor to start without the contractor having signed the contract? There is no written contract with signatures on it.

Dear Sir or Madam, I cannot even attempt to answer your question until you provide more details. For example: 1. Is there a signed contract? 2. What does the contract say about commencement of the work? 3. Is a person appointed to perform construction administration services on behalf of the Owner? This can be an architect or a project manager ... Typically the person performing CA work (managing the project on the Board's behalf) will issue a notice to proceed to the winning contractor. It typ says something like: "You are hereby authorized to proceed on the above referenced contract beginning _________. This contract calls for all work to be completed in ________ calendar days. Therefore, the contract completion date is established as ______________. Feel free to call or email if you have further questions. Have a great weekend!

It seems that perhaps the contractor is treating the initialed proposal as though it is a signed contract and is going to proceed with the work, while you expect to have a fully signed contract before the contractor begins. If the contractor has already begun work, someone with authority must tell him to stop immediately so that you can finalize the contract. I wouldn't assume it is anything but an innocent mistake on the contractor's part, but neither would I let him back onto the property without a satisfactory, signed agreement in place.