Well hello there, AlexanderSupertramp89!
Thanks for brining your question here to the Apple Support Communities! Congratulations on the new iPhone, by the way. The sole purpose of "Junk Mail" is so we don't have to read it, right? It's great to see that your Gmail and iCloud accounts have not been affected by this. With this information, we can conclude that the iPhone and it's software appear to be in working order. Here are two items to confirm:
Log into your Hotmail account using a computer web browser and confirm all notification settings. You may need to consult with Hotmail support for this. If all notifications appear to be set correctly, try the following:
If you still have issues with sending and receiving email, try removing your email account from your device and then add it back again:*
- On your computer, sign in to your email provider’s website to verify that all of your email is there or check if your email is saved somewhere other than your iOS device.
- On your iOS device, tap Settings > Mail, Contacts, Calendars.
- Tap the email account you want to remove.
- Tap Delete Account.
- Add your account again.
*iOS backups made in iCloud and iTunes include a backup of your mail settings, but not your emails. Emails deleted from your iOS device aren’t recoverable from an iCloud or iTunes backup. Learn more about backups in iCloud and iTunes.
Have a phenomenal day!