Yes. You can change your visa status from E-3 to any other nonimmigrant status while in the US.
Yes. As long as you qualify for H-1B, and there are visas available, you can change your status to H-1B. However, there is no legal obligation for you to switch to H-1B from E-3.
In certain circumstances you can travel to the US under the Visa Waiver Program to find a job or attend a job interview, but you cannot apply for a change of status to E-3 visa while you are on the Visa Waiver Program.
Yes. If you are already in the US in another nonimmigrant visa category, your US employer may file a change of status petition by filing a Form I-129 (Petition for a Nonimmigrant Worker) with the USCIS Service Center with Jurisdiction over the job location. However, you can only be approved for a change of status if your present visa category supports a change of status. For instance, you cannot receive a change of status from the VWP as this category does not allow a change of status.