If you are using an app to access your Office 365 account and it keeps asking you for your password it is usually because that app doesn't support 2-step verification. You'll need to either change apps, or use an app password. See more information in the "Setting up Office on your Mobile Device" section of the 2-step verification guide.

If it's an iOS device, try removing your account and re-adding it first. Apple's mail app needs to reconnect. If that doesn't work, your version of iOS may not support 2-step verification with the mail app.

In short it makes your account safer. 2-step verification largely eliminates spoofing and phishing as viable ways for other people to get into your account. By turning it on Graceland will be much less susceptible to data loss. We are also turning off POP3 and IMAP access to your email accounts as they are older less secure methods of accessing email that ignore 2-step verification requirements. You likely are using either of those.
Currently, when you log into your Office 365 account -- such as through Outlook or a web browser -- you are required to provide just your username (email address) and your password. With 2-step verification, you will provide a confirmation that you are logging in. That confirmation will come from a phone call, a text message, or an authentication app on a mobile device.
Yes. You will need to have a way for Office 365 to verify your identity separately from the machine you are logging in on. You will be able to set up multiple ways to confirm your log in in case you cannot use one of them. You can set up your office phone and two additional phones. You can also set up an authentication app on one or more mobile devices, such as a smartphone or tablet.
No. You will be required to confirm your log in whenever you first log in on a device, after you log out of your account, or every three months or so to re-verify your access on that device. Because there are multiple ways to access information in Office 365 you may have to answer multiple MFA prompts, especially on the first day. If you only ever access your Office 365 account on a single device you will rarely have to provide an MFA confirmation.

No. The goal of 2-step verification is to make it highly improbable for someone to pretend to be you not to make it impossible. It makes it hard enough that it is unlikely you will be breached. However, it is theoretically possible for someone to get access to one of your additional verification, such as by stealing a phone. Security researches have already proven that text messages can be intercepted or phones hijacked. However, 2-step verification by its nature is more secure than just a password.

Yes. Many other services, such as Amazon or Facebook also support 2-step verification, also known as 2-factor and multi-factor authentication. To find out if one of your other services uses it, and to get links to help set it up, go to the publicly maintained list at https://twofactorauth.org/ and search for the website or provider of the service you want to set up. It will list the types of factors that the provider supports (if any). If you see a check under "Software Token" they likely support an authenticator app just like Office 365 does (some companies have their own proprietary apps).