COS from H4 to H1 from USA

Hello Admin,

Here is my scenario, I am in USA on H1 visa from almost 1 years. Currently my husband is in India, one of the companies in USA has agreed to sponsor this H1 as he has 7 plus years of IT experience in core technology. He is currently serving the notice period in India and we are planning to apply his H4 (dependent visa) in end of May 2015.
He shall be travelling to USA by Mid of June 2015 on H4 visa for some personal event here in USA. Company has agreed to Sponsor/pay the fees for filling the H1.

So here are my questions:

Q1. If the application is picked in Lottery then who will be responsible for converting his visa from H4 to H1 in Oct 2015, please consider the following:
In April 2015 when the H1 will be filed he does not have any visa stamped on his passport
By Oct 2015 when the H4 to H1 conversion happens he will be already on H4 visa

Q2. If the Q1 answer is company, then may I know the approximate fees for carrying out he COS (Change of Status) from H4 to H1.

Q3. We may opt to change the employer by Oct 2015, in those scenarios what is the procedure for COS from H4 to H1 from our end. Can you please provide the list of forms to be filled and the approximate Fees.

It would be of great help if you could respond ASAP, as we shall get some time to think and take appropriate decision. Thanks in Advance.

– Rashmi