Question about Outlook Express

I use Outlook 2007 (non Exchange) and use it to manage my calendar and contacts (which i sync with my android phone without problem) and 4 different pop3 accounts which are filtered via rules into around a dozen different inbox folders. None of these are Microsoft email accounts. I have multiple accounts for the different companies I work with so cannot consolidate them into a single account. I use Pop3 because 2 of the 4 accounts have fairly strict server storage limits and also IMAP seemed to make a complete mess of outlook's interface creating entirely new personal folders for each one. I have spent years looking for a solution that would allow me to effectively take my desktop Outlook set-up with me on the move and more specifically:

1) Sync not just my calendar and contacts but all 4 of my email accounts with a cloud service so, when out and about, I can reference and search my historic desktop inbox and also get new email for all of the accounts much as i would with the Outlook 2007
2) Send emails through my four accounts without "on behalf of" or some other unprofessional looking addition
3) Sync all these remotely conducted incoming and outgoing emails back with the desktop when I return from being out and about
3) Sync any changes I make to calendars and contacts back with the desktop

I last investigated Outlook.com as a solution to this around a year ago and was given an unequivocal no. I also found no other mail service that would allow this without buying a dedicated Exchange server. It sounds like much has improved since then so...

Can Outlook.com now handle the above requirements?

If it can't do it as I describe above, is there an alternative approach e.g. using Outlook.com as the primary receiving app with Outlook 2007 some sort of syncing partner? I believe Gmail used to allow this but they now put this functionality behind a paywall.

Any guidance much appreciated!