Can I run my static website from an S3 Bucket, and add password protection?

This is possible using CloudFront and Lambda@Edge. See the answer here: https://stackoverflow.com/a/45971193/4550880

I think the AWS SDK for Javascript is what you're looking for. To be fair, it wasn't available when you posted this question 2 years ago. It allows you to login with Facebook, Google or Amazon. Here's another resource using AWS login.

AWS doesn't provide a way to do this directly right now. The S3auth solution you mentioned is nice in that your bucket/objects remain private so that a direct access to the bucket does not allow objects to be read without your private credentials. The disadvantage of the s3auth approach is that it relies on you trusting s3auth with your private credentials. If your credentials are compromised at any stage, it could be costly depending on how someone might abuse your access rights.

If you make your objects publicly readable (as you do when you create a website), anyone who learns/guesses/knows your objects names etc can access them. Or indeed if the bucket is readable, then all they need is the bucket name. There is no real way around this except by tightening the S3 access permissions.

If you only access your website from certain IP addresses, perhaps looking at Bucket Policies may help. Scroll down to Restricting Access to Specific IP Addresses. This is not a password but it does allow you to restrict where accesses can come from at least.

Another common technique for providing temporary access to objects is Query String Request Authentication. This does not however match your original requirement of password protecting your S3 bucket website.