Glad you were able to get a refund and now have it all working! I didn't realise you were original a non-Steam account and were trying to add the Steam version to it and as you found out, they're not compatible. What you did isn't actually that uncommon (bought again on Steam and gave the key away), although your friend may have a problem if they still play as I heard recently SE has started deactivating accounts where someone gave the key away (because it was sold to the original person, not who it ended up with). I have no idea how exactly they're doing it, I just remember seeing a post on their news pages about it. Your friend will also have a problem buying Heavensward because Steam shows it as a DLC so won't sell it without the base product so they'd either have to buy it again (and ignore that code) or get you/someone else with the game on Steam to buy the expansion and give them the code (and for future expansions too...)
To answer your questions:
The Steam and non-Steam versions of the game are actually exactly the same files on your disk, it's just the code you use to register the game flags your account as Steam or non-Steam. This makes no difference to how you install/play the game, only when it comes to expansions (as well as the Steam versions have the additional option of paying via Steam wallet if they want to). This is why you can log into the game using a Steam or non-Steam account and it doesn't stop you and it's also why you can log in on a friends computer, the game doesn't care how many PCs you install the game on, all it cares is about your account having a licence for the game and a current subscription (the game is useless without an account and each account has a code associated to it).
So basically since the game will always be in your Steam account now, you can either reinstall it from Steam (as it's not really Steam specific) or just get the launcher from the SE site and install it separately. The result will be the same.
Also just to clarify, the expansion comes, with normal updates, via the launcher as you've seen, there's nothing different to download. If your account you login with has the expansion, it'll offer to download it (and if you say no you simply won't be able to go into the new areas until you do).