I have been using Mozilla Thunderbird since 2005, when I imported the messages from Outlook Express, I have mails in it from around 2003. I have moved my Thunderbird profile directory (the one which contains the mail database files) several times over different operating systems, including Windows XP, Fedora Linux and Windows 7. I'm using 8 email accounts from 7 providers.
However, my first suggestion is to install it, then create one or two dummy email accounts, send some emails to them, and start experimenting with them.
The most important point to understand is that you can set up the account profiles in a way which removes the mail from the server (deletes it after it was downloaded). If you use that setting, you have to be absolutely clear, what are you doing, because *YOU CANNOT REPEAT THIS PROCESS* with your real accounts, and you might end up with a mess.
However, if done right, there should be no problem.
I've set up my Thunderbird in a way, that all incoming email comes into the same incoming folder, and I use filters to put the emails into different folders. For example, I'm subscribed to several different mailing lists, so I have folders for them, and the filter has a rule for identifying them by sender address.
But you can set up the structure by using separate incoming folders, as far as I know (see documentation about this).
I use the POP3 protocol to connect to the mail server, and when an email is downloaded, it is deleted from the server (there's an option for that), so all my email accounts are essentially empty most of the time. (I've chosen this option back then, because I never really understood, how does it work, if I let it in "not delete on server" mode, and sometimes, somehow, I ended up downloading the mails that I had already downloaded.) As a result, I have a 3.5 gigabyte Thunderbird folder, with all my mails in it.
I know that there's an another protocol, called IMAP, which makes it possible to have the emails on the server and in Thunderbird in a parallel way, but I haven't researched it to get a good understanding. You should.
Okay, and a word about sending mails. You can set up several different SMTP profiles, and join them to email account profiles in Thunderbird, so that when you send an email, you choose, which email address you are sending it from, and it will use the appropriate SMTP server accordingly.
So I have my Internet Service Provider's SMTP, and I have Google SMTP as well. Some SMTP servers allow sending emails with any "from" email address (the ISP's server is like that), some others allow only with their own.
That's it. Consult with the documentation pages, really. They are excellent.
Ah yes, about your older Sent-Mail. That's a problem. I'm not sure which is the right process to get them downloaded *into Thunderbird's sent-mail directory*. I'm not saying that it is necessary, but if you have the idea to get everything downloaded on the local machine, then you have to figure this out.
Again, you can keep your emails on the web servers, but if you decide to keep them only on your local machine, be sure to have a regular back-up practice in place. It would be a shame if you lost some mail because of a hard drive failure.