Is the author obliged to answer all emailed questions regarding his or her paper after it had been published?
From a formal or legal perspective, there is no requirement that an author do anything: you can go full J.D. Salinger and vanish into a hole if you want.

From the perspective of scientific ethics and custom, however, an author is expected to share sufficient information in order to support their colleagues in understanding and building upon their work.  This does not mean they need to answer any and all questions, however, but only those that are a reasonable part of the process of scientific information exchange.  

Among questions that a scientist should answer would be help with interpretation, methods, and replication (e.g., "I can't track down the reagents you used: can you please tell me where you got them?").  Just like on StackExchange sites, it is reasonable to expect the asker to show evidence that they have worked to understand on their own before asking.

Among questions that a scientist a not obligated to answer would be:


Requests for pre-publication information (e.g., "Can you please give me your next unpublished dataset so that I can try to scoop you?")
Requests for remedial education on a subject (e.g., "You wrote a paper about protein evolution, so if people evolved from monkeys why are there still monkeys?")
Requests for personal or otherwise scientifically irrelevant information (e.g., "Do you have a crush on any of your co-authors?")
Straight-up craziness (e.g., "Given your expertise in artificial intelligence, will you please help my trash-can robot learn to irrigate the Sahara Desert from Archie Bunker?" [this one is from personal experience])


Now, many scientists are generous with information and will still help with remedial questions, share pre-publication information with people they trust, etc.  I would consider many of these things a matter of ethics on a broader stage and of good citizenship, but not so much something that colleagues are likely to look down on you for not doing.