How to log into several Outlook accounts online at the same time

Is there a way to quickly merge my Google accounts

No. I actually emailed Google about this one after migrating my domain email to Google Apps, since I'd had another Google account set up. They don't offer any services for merging Google Accounts.

let me easily check both of my Gmail accounts, all from one place, whilst still keeping them relatively separate

Is Google's own Multiple Sign-In enough?

My Hotmail account, what can I do about that?

Sadly, not much. Start the painful process of switching those accounts on various sites over to one of your new accounts. Crap like this is the #1 reason I bought my own domain: So my email address would never have to change again.

I downloaded external mail software (Outlook) where I could easily connect to both my email addresses and move email from one account to the other by drag & dropping the folders/emails.