The police can't help, they are too clever.
Complete BS. Cable/phone companies have logs about what each employee does. They always leave traces behind which the police can use to find them.
But if you work for the cable company and the phone company, you have access to much info that you can misuse.
Sure, but they don't have access to your passwords or system, unless you use weak passwords and don't protect your system.
I want to know if Avast can stop anyone from logging in from another IP. Can that be done?
Tools can help with security, but it always starts with the user and the user will always be the weakest link.
3-, 4-, and 5-letter words, plus caps and numbers
Words are never strong. All words are in dictionaries that anyone can use.

If you suspect they are accessing your system remotely, start with checking for software that can be used for remote access on your system and remove it (or at least disable it and block it in the firewall)