tech support scams

Looks like typical scam. Never let someone to access your Mac remotely.

http://www.thesafemac.com/tech-support-scam-pop-ups/

Download and run Malwarebytes Anti-Malware: http://www.adwaremedic.com/index.php

You can also remove the adware manually using the steps provided by Apple: https://support.apple.com/en-us/HT203987

After that, make sure your default search engine is set to the one you always use. To do so, Open Safari - Preferences - Search.

Go to Safari menu (At the very top right side of your screen next to Apple icon), choose "Quit Safari"

press "Shift" button and while holding this button on your keyboard single click on the Safari icon on your Dock.

Open Safari - Preferences - Privacy - Remove All Website Data.

Also, I believe it would be a good idea to install AdBlock.

If anyone calls asking to repair your computer hang up, do not engage them for any reason.

If you would like an analysis of your computer you may post an etrecheck report and the volunteers here will make suggestions based on this report to assist you. No one here will ask you for anything in return for providing this service.

if you think your Apple ID has been compromised see this article:

https://support.apple.com/en-us/HT204145

You may wish to change your Apple ID log/pass with a password you do not use for any other on line service. If you use your same log/pass combo with another service (e.g. Yahoo, Twitter, Facebook, etc.) and that service is "hacked" then the perpetrators have to only try typing this same combo in any other service to get into your other accounts.

Change your Apple ID password - Apple Support

https://support.apple.com/en-ca/KM205079

Change the Email associated with the ID if you wish to remove the email in use:

https://support.apple.com/en-us/HT202667

Also consider enabling two factor authentication if you have not done so already.

Availability of two-factor authentication for Apple ID - Apple Support

See this link regarding you security questions.

https://support.apple.com/en-us/HT201363

That is definitely a scam. I just paid only $200+ to have the battery replaced in my wife's MacBook Pro, a job which required replacing the whole top case, which is not a quick task.

Nobody but a scammer will charge you $700 for a software fix.

Apple will never call you unless you request a call back or set one up