Transparent tweets haven't actually been saved by anyone. They were automatically added during the saving of a direct reply because it's nice to keep context. They will appear normally if they later receive an actual save.
When a tweet is shown in a larger box it means that multiple people saved it.
Yes. All saved tweets are permanent and will always appear under the tweeter's username at http://tweetsave.com/TwitterHandle. Individual saved tweets will always be available under http://tweetsave.com/TwitterHandle/status/TweetID.
The link to the "My Saves" page is indeed based on a cookie, and if you change computers you will have a new save history. You can still view your old list if you remember the link. You can even continue to use it as normal if you edit your savefolder cookie to reflect your old save folder name (so write it down if it is important).
If you want, you and a group of friends can even share the same save folder and name it as you wish. The only limit is that the folder name must be alphanumeric (plus _ and -) and no longer than six characters. How: simply edit the savefolder cookie and tell them to do the same. After you have done this and made some saves you can view your collective efforts at http://tweetsave.com/saves/Folder (replace the folder name).
The old version of the addon sometimes causes a screwy twitter interface. You can refresh the whole page to fix it but you should update to the last version. See the addon link at the top of the page.