Do email clients like Thunderbird download the mail to the PC they're installed on, or do they access it in real time?
If I receive mails with attachments, are they being saved on my PC even when I don't open them?
If you want to leave your messages on the server you should not have "For at most 14 days" checked.
Theoretically these two options (For at most 14 days / until I delete them) counter-act each other (one is saying delete, one is saying don't delete) and should really be seperated by a radio checkbox rather than tick boxes. I'm not sure how Thunderbird actually reads these options (ie, which one overrides the other) as I haven't got a copy of the source code but by what you are describing, the "For at most 14 days" checkbox is probably taking priority.
You should only tick "Until I delete them" if you want all your messages to stay on the server.
Please note that depending on how much space your email host gives you, this could rapidly run out if you do not download and delete messages from the server.
The attachments you are "downloading" are already downloaded, you're simply saving them from Thunderbird's storage to where ever you are asking it to save to.
Of course, there are other possible reasons your webmail disappeared but without more information such as how long you've been using Thunderbird, whether webmail did a maintenance purge, etc etc, it's hard to say for sure.
You can't really put the email back on the server, it's technically possible server-side but nothing you can do as a user, unless you forward all 700+ emails back to yourself. I'm not sure why you'd want them all to stay on the server anyway, if you want to save important emails or documents you should download them and save to an off-site backup like DropBox or sign up to Gmail which has so much storage you don't have to delete anything.
(Other off-site backup services and web mail services are available)
Since you are using POP3, yes, the messages get downloaded from the server to your local computer. Attachments are technically part of the body of the message. Email was designed 40 years ago and there was no support for attachments so they found a way to 'encode' the attachments into the body and email clients 'parse out' the attachments for your convenience.
So, when you download the body of a message, you are downloading all of the attachments, even when you don't open them.
If you don't like the idea of leaving something on your computer (say you're checking your personal email account while at work) you might try a portable email client like TrulyMail Portable or Thunderbird Portable. Both allow you to run the email client from a USB drive and keep your messages in your pocket.
Put simply, yes and no. Depending on how your email is configured. There's two possible ways:
Attachments: No, those are downloaded from the server when you click on them.
Most of the time you can change how they behave but you'd have to Google it to find out how.