What you have described suggests to me that you're using POP, and apart from the Inbox, POP does not expose folders on the server to the mail client. Furthermore, there is no support in POP for the client to write to the server. The client can get messages and it can delete them. It cannot see nor write to the Sent folder.

For what you want to do, you need to be using IMAP. Then all the folders on the server will be visible and accessible in your mail client, including "Sent" .

Set up the account in Thunderbird again, but when it offers a choice between POP and IMAP, choose IMAP this time. If it doesn't offer this choice, you need to do some research, find the IMAP settings for your account, and use the manual configuration option to apply these.

When the Sent folder on the server becomes visible, you'll be able to move or copy the contents of the local Sent folder to it. I suggest you select the messages, then use right click and "Copy to" or "Move to".

The problem is that the mails are not saved on the associated mail server (in this case, pop.mail.yahoo.com),

With a POP account this is the expected behavior. Only the Inbox folder is visible on the server, all other folders are local. https://support.mozilla.org/en-US/kb/glossary-terms-including-types-accounts#w_pop

In order to get access to other folders on the server, e.g. your Sent folder, you'd need to set up your account as an IMAP account. https://support.mozilla.org/en-US/kb/glossary-terms-including-types-accounts#w_imap

thus each time I have to reinstall the software I lose all my outgoing emails.

You should not need to reinstall the software in the first place. And even if you do so you'll not loose any mail, unless you deliberately delete your Thunderbird profile. With POP all mail is local on your computer, including the Sent folder. http://kb.mozillazine.org/Profile_folder_-_Thunderbird

if there is a way to bypass the issue by backing up the 'sent' file located in the Thunderbird's Mail folder located on my HDD.

Actually you should backup the entire profile. http://kb.mozillazine.org/Thunderbird_:_FAQs_:_Backing_Up_and_Restoring

Is it possible, after reinstalling the application, to paste the old Sent file in the Mail folder?

As said before, nothing gets lost when reinstalling the application. If you want to move to a new computer you simply move the profile. http://kb.mozillazine.org/Move_to_a_new_PC

do I have to backup the file 'Sent.msf' as well?

If you want to backup individual mail files you don't need to backup the *.msf files. They are just index files and do not contain mail. They are rebuild automatically when Thunderbird is started. In any case, backing up the entire profile is the recommended way.

thank you!