Emails I send and recive are not going through to me or the other person.

For Outlook.com/Hotmail, the most relevant checks are these:

1. Check the Other tab, Junk Email folder, and Junk Email settings. Important messages such as app codes can be filtered away from the main inbox.
2. Check whether messages are just sorted oddly. In Outlook.com, select Filter > Sort > Date to restore chronological order.
3. Check whether email forwarding is enabled.
4. Check whether any email rules are moving or hiding messages.
5. If a specific sender’s messages are not arriving and there is no visible error, ask that sender to review Sender Support in Outlook.com.
6. If sending fails, check whether:
   • the message has an attachment larger than 25 MB,
   • there is a non-delivery report (NDR) explaining the failure,
   • the recipient address has a typo,
   • the recipient mailbox is full,
   • the recipient server has issues,
   • the message was blocked by the recipient’s spam filters.
7. If a message is in Sent Items but never reaches the recipient, try sending a simple test message with:
   • no attachment,
   • no signature,
   • one recipient only. If signatures are enabled, remove the signature and send again.
8. If only some senders cannot reach the mailbox, the problem can be on the sender’s side. A sender’s mail server or IP reputation can cause Microsoft to reject their messages even while other mail still arrives.
9. Check the account itself. If the password was changed recently, update it. Also check whether the account was temporarily blocked because of unusual sign-in activity.
10. If sign-in problems are involved, use the Microsoft account sign-in helper.

If none of the above resolves it, use Outlook.com support from the Help menu, enter the issue, then go to Still need help? and select Yes.

References:

• Can't send or receive email in Outlook.com | Microsoft Support
• Sent email in Outlook.com comes back "delivery failed" | Microsoft Support
• Not receiving some hotmail emails - Microsoft Q&A