SQL Server Developer Edition is a free download starting with SQL Server 2014.
You can download SQL Server 2017 Developer Edition for Windows, Linux, and Docker from the Microsoft SQL Server downloads page.
SQL Server 2017 Developer is a full-featured free edition, licensed for use as a development and test database in a non-production environment.
We can't advise on detailed licensing issues, that is something you should read about on the MSDN Subscriptions site, and contact support if necessary. With that said, I can respond to other points in your question:
- Next to each download it says: "No product key is required". I assume this means that a 28 day trial is initiated after download?
You have a choice here. During installation, you are presented with a screen like:
If you choose the pre-populated product key, you are installing Developer Edition (not a trial) under the MSDN Subscriber licence terms. No product key is required for this option, because one is provided for you.
If you choose Evaluation, you are installing a time-limited trial of Enterprise Edition, with different licensing terms. This is just the same as if you had download Trial Edition from anywhere.
- What is the difference between SQL Server Developer; SQL Server Web Edition and SQL Server Business Intelligence? I assume that SQL Developer can do everything that SQL Server Business Intelligence and SQL Server Web Edition can do?
SQL Server Developer contains all the features of SQL Server Enterprise Edition, just on different licensing terms. You can compare the various editions here:
SQL Server Editions
You cannot download software from MSDN Subscriber Downloads without a Microsoft account associated with a valid subscription. As it says on the Using Subscriber Downloads help page:
You must be signed in to your Microsoft account that is associated with your subscription in order to download products on Subscriber Downloads. If you are not signed in, you'll be prompted to sign in when you attempt to download a product or view product keys. Everyone, including non-subscribers, can view Subscriber Downloads, but only active subscribers can download products and keys.