Hi,
1. Good news that both providers are using cpanel. you will be familiar with the new environment.
2. Great. That means you can just download/ftp all the html files from old provider and upload into the new provider
3. When you change hosting, if you change the nameservers to the new provider, the MX for the world will be automatically pointed to the new provider. For getting the emails from other domains in your old provider, its better if you change the MX entry in the old provider and point to your new provider, or ask them to delete your domain/data from there.
4. Yes, You will need to create the email-access again in the new webhosting provider.
Hope this helps.