Hi. thanks for being the first support question.
This first release of the plugin is aimed at people that are familiar with Gettext files. I intend to write some thorough instructions as soon as I can, but for now my priority is getting it working well.
In answer to your questions:
POT file is optional if you're just using this plugin to do your translations. It's there for the people who will want to use it. If you don't know what it's for then work without it.
You can export the files directly. just click Download in the editor to save a PO file to your local computer.
For the plugin to be able to create and save files to your web server, the web server needs write access. If you don't have permission, or don't know how to do this, then you will be restricted to downloading files and then copying/FTPing them into your server.
If you can't save the files directly to your server, then the plugin won't be able to compile the MO files either. WordPress reads the MO files when actually looking up translations.