Generally, in the systems that allow you to have multiple accounts, messages that are sent in reply to something carry as the "From:" address the "To:" in the message you're replying to. This is particularly handy if you use the unified inbox feature (I don't, as I prefer to know exactly what account [and it's corresponding inbox] I'm looking at at all times). You can usually set the default account you wish mail to be sent from when composing a new message when viewing the unified inbox and I think some will ask you which account you want to use as your "From:" as well when that would not be clear.
If you are sitting in the inbox for a given account the "From:" address defaults to that account unless you've used a mechanism in the client to override that.
You can definitely send mail from any one of the accounts you have included and have the e-mail address that's associated with that account as the "From:" in these interfaces.
I have never used an online interface that doesn't handle folder management on an account by account level for IMAP folders. There's no real equivalent of a "local folder" in this case, as everything is "in the cloud"/"on the server" and you're just looking at it all through a web interface.
Remember if you go with Gmail or Outlook to look at their privacy statements if that matters to you or the business you're doing. Google pretty much says that "your e-mails are all open for us to scan" with the goal of coming up with targeted advertising for you and/or other data mining. No one reads them (as in no human person) but they're all metaphorical open books. I don't know what Microsoft/Outlook.com does in this regard. Any web-based service would have to be granted permission to access your e-mail, as they can't present it to you otherwise, but it's what they can do with the data presented behind the scenes that can differ significantly.
Brian AKA Bri the Tech Guy (website in my user profile) - Windows 10 Home, 64-Bit, Version 1803, Build 17134
In a modern society where everyone thinks their opinion deserves to be heard nothing annoys me more than individuals who mistake their personal preferences for fact.
~ Commenter TheCruyffGurn on the The Guardian website, 8/13/2014