Yes to most of what you said hehe,
You can use the clients for almost any email account, and you set it up (yes the POP3 means it's accessing the actual account online with the ID & PW as well as the incoming and outgoing settings which are usually provided by your email account company. Then you decide if you want the emails to stay on the webserver or not (usually not a good idea since they will end up piling up in your inbox forever that way lol) which I usually tell it not to leave them on the server when I download them to the client.
When you set these email addresses up you also setup the outgoing server information in the client also, this allows you to send directly from those addresses also. What you'll then have is usually a drop down menu of email addresses to choose from (if you have multiple ones in the client that is) and you tell it which one to send from and away you go!
Having a webmail account depends on if it "is" a webmail account or includes webmail setup or not. What I mean by this is, we all know how Hotmail, Yahoo, Gmail are all webmail accounts right? Well your ISP might give you email (today it's almost all got webmail of course), but years ago there was no webmail for a lot of internet providers, you set it up in a client and that was it. Today most include webmail also so you can do it either way your own preference is. Having said that, when you have a hosting package you almost always have a webmail system built into it so you can do email that way, but you can also find the settings in your cpanel emails (usually there is a link on your email accounts listing to get your email client settings) and then set it up in a client program.
Email is then sent automatically "from" your server if you set it up to do so, and it's also incoming from your server as if you were checking it on the webmail. What it won't do is archive your sent list because your email program is doing that for you. There may be settings on the webmail that will do this, but I haven't checked since I don't really ever use the webmail part of any of my emails. Gmail "does" archive all your incoming emails for you though if you tell it to do so.
Just tell the server not to keep them on the server when you check it and you're all set to go. Hope that helps and let me know if not