Microsoft has been killing off Hotmail little by little.
No, they rebranded it to outlook.com. Hotmail addresses going back to 1997 will still login and still receive mail perfectly fine. You can still create an @hotmail address up to today.
From accounts that send a lot of E-mail, webmail (including gmail) usually results in message deferral which makes it take much longer to arrive. The idea is to "throttle" possible spammers, since generally it is spammers sending out a crapload of E-mails. It's worth trying another webmail though as it might function differently.
Just "anecdotally" but we've found gmail to be the biggest pain in the *censored* when it comes to sending E-mails. inevitably complaints that customers didn't get monthly statements or reservation notifications turns out to be because gmail is involved somewhere in the loop.