If someone got onto my computer remotely and installed malware/remote-access malware so they can easily get on in the future, if I do a fresh install with a new downloaded .iso, will that get rid of any/all malware without question?
The short answer is Yes, it will get rid of all software, good and bad, if you wipe the hard disk and do a fresh install of Ubuntu. And you don't have to download a new .iso file, either. Even if the malware found its way to the MBR, it will be overwritten. This applies whether the malicious software is on Windows or Ubuntu or both.