This is is apple real mail?

No, but you can forward Apple that mail. Check this article:

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Scammers try to copy email and text messages from legitimate companies to trick you into entering personal information and passwords. Never follow links or open attachments in suspicious or unsolicited messages. If you need to change or update personal information, contact the company directly.

These signs can help you identify phishing scams:

• The sender’s email address or phone number doesn’t match the name of the company that it claims to be from.
• Your email address or phone number is different from the one that you gave that company.
• The message starts with a generic greeting, like “Dear customer.” Most legitimate companies will include your name in their messages to you.
• A link appears to be legitimate but takes you to a website whose URL doesn’t match the address of the company’s website.*
• The message looks significantly different from other messages that you’ve received from the company.
• The message requests personal information, like a credit card number or account password.
• The message is unsolicited and contains an attachment.

Report phishing attempts and other suspicious messages to Apple

To report a suspicious email, forward the message to Apple with complete header information. To forward the email: In macOS Mail, select the email and choose Forward As Attachment from the Message menu at the top of your computer screen.These email addresses are monitored by Apple, but you might not receive a reply to your report.

• If you receive what you believe to be a phishing email that's designed to look like it’s from Apple, please send it to reportphishing@apple.com.
• To report spam or other suspicious emails that you receive in your iCloud.com, me.com, or mac.com Inbox, please send them to abuse@icloud.com.
• To report spam or other suspicious messages that you receive through iMessage, tap Report Junk under the message.

*To confirm the destination of a link on your Mac, hover your pointer over the link to see the URL in the status bar. If you can't see the status bar in Safari, choose View>Show Status Bar. On your iOS device, touch and hold the link.

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from: Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support

Yes, it’s referring to this policy (change the country as appropriate with the icon in the top right)

iPhone Out-of-Warranty Battery Replacement Credit - Official Apple Support

You will need to visit an Apple Store or the AASP who did your battery replacement to get the credit.

I responded to a post yesterday about this. It appears that it is a valid Apple address and they are notifying people that had batteries replaced before they reduced the prices and they are trying to work and refund these individuals.

I see the @insideapple.apple.com suffix all the time - Apple emails about news@..., trade in offers/ads, satisfaction surveys after recent support contact, etc.

Yes, it is a real email.

Any emails from Apple will never ask for passwords or credit card info.
The from address will always be apple.com (as this one shows).

Yes, it is.

(162192)

So I stand corrected 😉

But @inside.apple.com seemed suspicious to me.

Don't feel bad. I did the same thing yesterday, and was shown the correction.