1) is there an API method to capture/store the thumbnail image of the YouTube video at 3:59 which is not the autogenerated thumbnail or the cover thumbnail uploaded by origi
1) It's not possible with the YouTube API V3, you can only get the public thumbnail of the video.
2) If you want to do that, the only way is to :
I hope this helps
Edit
Edit about the review agreement
Seriously who reads the terms and review agreement ?
All custom thumbnail images must follow our Community Guidelines. Thumbnails that are sexually suggestive, violent, or graphic may cause your video to be age-restricted. Age-restricted partner videos will not have advertising served against them and will not appear on browse pages.
It's not forbidden to create image and put her on a video to attract users. All you need to do, is to keep the meaning between the thumbnails and the video.
I see you want to use something like a screenshot of the video so you respect the connection between the thumbnails and the video !
The content owners of the video cannot be disagree with that.