MacBooK hacked or gmail compromised, or both?

Is my loyal old MacBook being invaded or is it me?,

5 IP addresses activity sign ins to gmail from iphone from NSW, Im in Qld. Added tough passwords and 2-step process, but emails recieved by friends, one with codes, one about holiday, not mine. So it's bypassing this verification process

I would like to delete gmail accounts, but all these are my name and like Tina Turner says, 'I'd like to keep my name'.

Running AVAST, and a few warnings, no infections...

Not a techie, so any advice in slow short words is welcome!

E