Yes, it is possible. After you upload/submit the video you can choose autogenerated thumbnail images for your video:
or you can set your own image after your account gets verified:
The verification process looks like this:
If the video is not yours you can still do it by editing the embed code - for example:
official embed code:
<iframe width="560" ...
By default sharing a folder in Google Drive (in your case "Big Folder") - any file or folder inside it (in your case "Small Folder") is shared with the same people.
Therefore, if you have not changed settings, everything you move from "Big Folder" to "Small Folder" will be shared the same. And no one will lose access.
But if you have changed the sharing ...