I have trouble trying to follow along with some of your questions, but I'll see if I can give at least a few answers...
1. If your account is public, everyone has access to it. They can view, save, and use any pictures that are in your account. They can't make any changes to your account, though, of course.
2. If your account is private, then people can't access your account. However, if you post a picture, people can then view, save, and use that one picture. This doesn't give them access to your other pictures, though.
3. On the other hand...if someone can successfully guess the name of one of your pictures, they will be able to view it. For example...
Without even going to your account, someone could guess that you have a
ChickenLittle.jpg and they would get the above picture. Someone could guess that you have a
NakedJP001.jpg in one of your private folders (you can have a public account, but make areas of it private). If they're wrong, nothing will happen (aside from an error page). If they're right...well, it wouldn't be pretty. And the
001 implies that there is a series of such pictures, so they could try looking for
NakedJP002,
NakedJP003, etc.
Keep in mind that isn't normally something you have to worry about. I mean, why would someone go through the effort? It's possible to access your private pictures, but there's a lot of hassle involved. However, if you have any pictures that you absolutely don't want people to see, don't give them obvious names.
4. Although you have some rights over some of the pictures in your account, this doesn't mean that other people aren't allowed to use them as they please. This may be different for private pictures (although PhotoBucket is allowed to do what they please with these pictures), but all public pictures are basically up for grabs. Perhaps you should read the license agreement from the Terms of Service:
Photobucket does not claim any ownership rights in any User Content that you choose to post to the Site. After posting User Content to the Site,you continue to retain all ownership or license rights in your User Content and you continue to have the right to use your User Content as you did prior to such posting. However, by posting or making User Content available through the Site or via the Services, you hereby grant to Photobucket a nonexclusive, royalty-free, transferable, worldwide license to use, copy, modify, prepare derivative works from, distribute, publicly display and publicly perform (whether by means of a digital audio transmission or otherwise) and process your User Content, or any part of it, solely on and through the Site and Services, including without limitation (a) adapting the format of your User Content (for example by encoding or transcoding) for suitable display on the Site; and (b) displaying, in Photobucket's sole discretion, your public User Content in search results generated by the Photobucket search engine. In addition, where you have made your User Content public, posted a link to your User Content on another website or otherwise shared a link to your User Content, you grant to Photobucket a nonexclusive, royalty-free, worldwide right to sublicense the right to copy, modify, prepare derivative works from, and distribute your User Content as necessary to perform the Services, including without limitation the printing services offered by Partner Sites. In connection with the above license, you provide your consent for Photobucket personnel, including Photobucket contractors and service providers, to view your User Content at any time for the purpose of providing the Services and filtering content that violates this Agreement. You acknowledge that you are solely responsible for all Content you submit to the Site or provide to the Service. You represent that you either own the User Content or have the rights necessary to grant Photobucket this license.
Your license to Photobucket with respect to any particular piece of User Content will terminate once you or Photobucket remove it from the Site.
Basically, we can do whatever we want with your pictures. I can put the pictures on my own account and display them wherever I wish.
5. Right-clicking on a private picture gives all of the same information as a public picture. There's no special encoding/encryption. Basically, the only thing that is truly private is the account/folder itself. Somebody would need your password to get into your private account. But remember,
anyone can view a public account.
6. People absolutely can not trace a picture back to your computer. Period. There's no way to get access to anything from your computer if you post pictures from PhotoBucket.
Phew. I'm sure I've probably created more questions for you, but I hope I have at least provided some answers here.