1. If you create a regular sales order that contains a subscription product, shouldn't it then create a new subscription? I'm not able to, but it may be that I'm doing something wrong.

Quotation Template field should be added in quotation upon installing `website_contract` (the field comes from `website_quote` which `website_contract` depends on).  To this field, you need to assign a quotation template which is linked to a subscription template in order to generate a subscription upon confirming the quotation.  Odoo proposes "Default Template" to the Quotation Template field of quotation by default.

2. If you have several subscription templates (for monthly and yearly subscription for example), is it the tag that is used to group them together in the website (where the user can change plans)?

Yes, the tag is used for this purpose.

3. Is it possible to sell subscription products in the ecommerce?

Yes, it is possible.  Just publish the subscription product on the website as normal.