Can a CloudFront origin be another CloudFront distribution?

If you just want to call the API with multiple names, there is no need for second distribution but you could add alternative names to your "api" distribution. https://docs.aws.amazon.com/AmazonCloudFront/latest/DeveloperGuide/CNAMEs.html

I'm not sure how good idea is to have 2 levels of caching but you could configure "api" distribution as custom origin for "project" distributions. I recall there might be some loop-detection in Cloudfront not allowing this, but should be able to work-a-round by limiting origin request headers you forward from "project" distribution to "api" distribution.

Yes you can chain CloudFront distributions, but only 2-deep. It's not optimal however customers have various use-cases to do this.

If you try 3-deep it will fail.