Lets take em 1 at a time
Server admins can install a plugin that will log the steamid/ip/hwid, and this plugin can then crosscheck if any of them has a ban on them, so it might not seem automatic but in most cases it is, its just not checking all the time and steam has a max connection it can search pr hour so ban evasions can take a few hours to kick in.
If you buy a new dayz account on your gf pc, and then login to your own pc where you used the old acc, the new account will be flagged and most likely get banned. Now that said, if you make a real clean of your pc and dont login the any banned accounts so that after you make the new account on your gf pc and only login with this new account on your own pc(its importent you have never logged any other steam acc in on this windows install) theres a good chance it wont get banned(but there is a chance depending on what they logged)
The best way is to make the new acc on your gf pc, then install windows fresh on your own pc, get a vpn/new ip´ you should be good to go. A hwid spoofer will make it ever more safe but might not be a requirement.
Regarding your router question, this is more tricky to answere as it depends on your isp, some isp assign a ip to your model thats build in the wall of your house and you cant just change that, and other assign the ip to the router, i would contact your isp and ask them if getting a new router would get you a new ip, if not you should look into a good stable vpn, you can get a stable vpn for around 10$ a month at most.
dont worry about dynamic ip's as its less and less used externaly, most isp that tell you that you have an dynamic ip its really only dynapic in the server cluster your connected to (within the isp's server farm) and not an external dynamic ip, so the ip the game server see is static even when your ip has changed, this is why a vpn is the best way to change your ip.
hope this helps.