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I'm not entirely sure I understand all of your question. If you'd like to use Outlook Web App to access your email in Office 365, you can easily do that. You don't need to use Outlook at all if you don't want to.
In Office 365, you have user accounts and licenses. Let's discuss them briefly
You can have as many user accounts as you'd like, and they are just accounts you can use to sign into Office 365 to access your services. You don't pay for user accounts at all, and you can give a user account administrator access so they can administer your company without licensing that user.
Licenses, on the other hand, are what you're purchasing when you purchase your Office 365 subscription, and they're frequently referred to as seats. If you want to use any of the Office 365 services, like Exchange Online, Sharepoint Online, or Lync Online, you need a corresponding license to do so. After you purchase a license (or receive temporary licenses as part of your trial) you can apply that license to a user account to give that user access to those services.
It sounds like what you want to do is gather together all of your email addresses into one place. When you assign a user account an Exchange Online license, that creates a single Exchange Online mailbox for that user. You can create several email addresses for that mailbox, which we call aliases, and any email you send to those email addresses will wind up in that one central mailbox. You can then setup forwarding from your external mailboxes, or use the Connected Accounts feature to pull email from those external mailboxes.
In Outlook, you can open several mailboxes at once. It sounds like that's what you're used to doing, as you want to use several different boxes from several different domains in one account. In Exchange Online you'll only have the one mailbox and you can then put all of your mail into it. In the case of the setup that it sounds like you had previously, you didn't actually have one mailbox account, but actually several. You then would've used Outlook to control a single Outlook profile that gathered email from all of your accounts.
Please let me know if you have any additional questions and I'll be happy to help.