1) So, the email accounts you mention as ‘Online’ have two basic differences from the ‘offline’ ones of thunderbird and outlook, etc. The first is that those supply you with an actual e-mail address, xxxx@hotmail.com, or whichever. Thunderbird does not, and I’m not sure about outlook but i believe it doesn’t. The ‘offline’ programs are for getting, sorting, and otherwise dealing with your mail. They usually still have to work through another account, since you can’t get a xxxx@thunderbird.com account (at least, not without paying money).
The second difference is why they’re called online and offline. Online email are web sites, and all information, except some temp files, are online on the servers of the respective service. Hence storage limits on these accounts: you are using their storage, not your own. The Offline programs get your mail from these servers, and then store them on your computer (or flash drive), also called ‘stored locally’. So no, these programs interact with the online versions, but they are all running on your computer, and can run offline (you can’t get new mail while offline, but you can otherwise manipulate old mail), and store all data on your computer (or drive). This is why they might be good for you, since they allow you to back up that data, while online sites rarely have the option (though they tend to do their own backups).
2) ‘execute a program’ is simply another way to say ‘run a program’. Execute is just what the computer tends to call it.
Also, the portable apps thing @jerv mentioned may be blocked by a public computer, but it has the best chance of working, of all the options.