Set up an email for each subscription. E.g. yourname+nyt@gmail.com. Block anything coming without a + extension. When you want to unsubscribe, create a rule to delete any email from the specific address you set up for that subscription.
Not really - some webmail clients act like they have one but they almost never works.
The other way is when you create an account somewhere, first thing you have to do is to go in their privacy and / or notifications settings. Stop it at the source.