I fear this option is more expansive.
It is not exactly right. AzureWebsites offer in 3 modes - Free/Shared/Standard. Based on the mode you run, there will be pricing. If your application got limited visitors you can run on Free mode, which is least expensive.
First question : Is there any other option I didn't see ?
Not that I can think of. Your options are - OnPremises, Cloud, Shared Hosting Providers. You are exploring Cloud and Shared hosting providers.
Second question : SQL Azure and SQL Server are they exactly the same thing ?
SQL Azure is not exactly the same SQL Server. It got its own limitations -
- Transact SQL Limitations - http://msdn.microsoft.com/en-us/library/ee336253.aspx
- Other Guidelines and Limitations - http://msdn.microsoft.com/en-us/library/ff394102.aspx
- This thread showed lot of limitations and associated resources
Can I still use the Entity Framework the same way ?
SQL Azure got EF Compatibility - http://www.windowsazure.com/EN-US/develop/net/how-to-guides/sql-database/