No, these emails forged. They are not coming from Hotmail's server. I assume you're looking at the "From" header in the email message. Look at the MAIL FROM value in the SMTP envelope instead. Most likely that will not say Hotmail.com. Check https://www.xeams.com/difference-envelope-header.htm to see the difference between email header and smtp envelope.

I can confidently say this is a forgery because the example you sent is a common method of deceiving the recipient to think the message came from Hotmail. In fact, Xeams has a filter that looks for such tricks and assigns a score.