All you need to install your e-Sword module is to enter your Email address and "Product Key" you received when you purchased the module. If you do not have that information, just click on the "Recover Product Key" button on the dialog that prompts for this information.
If you have updated from a version older than 9.0 then you'll need to redownload all previously installed modules as the old ones are not compatible with the new version. You can redownload purchased modules using the original information you were given when previously purchased.
Topic Notes are now viewed within the new Reference Library. You can select this from the e-Sword "Tools" menu, or click on the far-right button on the main toolbar.
Use the Download menu from *within* e-Sword. Check out the "Download" demo on the Training page.
If you are on a home network, then run the "Internet Options" applet in Control Panel. Click on the "Connections" tab, then click on the "LAN Settings" button. ONLY "Automatically detect settings" should be checked.
If you are on a work network, then select "Options, Proxy Settings" from the Downloader menu and enter your network requirements. (Your IT department will have that information.)
If you are still experiencing trouble, then it sounds like either an Internet filter or firewall (e.g. ZoneAlarm, etc.) is blocking e-Sword from accessing its server. You can temporarily disable these and see if that helps, and if so then you can add an exception for e-Sword to their settings.
You can delete a resource by selecting "Options, Resource" from the e-Sword menu. Right-click over the resource you want to remove, then choose the "Delete" popup menu item. Confirm, and it is gone.
Check out the "Views" demo on the Training page.
Those are Bookmarks. Right-click on one to set it to the currently active verse; left-click on one to go to its reference.
Copy the e-Sword *program* directory (typically "C:\Program Files (x86)\e-Sword\") onto the new PC as this will get your various modules transferred. You will still need to install e-Sword from its setup program to properly install the various system files, etc.
Also, you will want to copy the contents of your e-Sword *user* directory (typically "Libraries\Documents\My Documents\e-Sword\") to get your personal notes and other customizations transferred.