Is there a shell extension or something that allows me to delete files from Explorer, even if it has to take ownership and adjust the permissions of the files first?
Take Ownership is a Registry file that adds an Explorer context-menu item to change ownership of a file.
For some other tools, and a caveat, see 6 Tools To Take Ownership and Get Full Control Of Files and Folders.
CrystalRich LockHunter will unlock or delete files locked by a process, though it won't change ownership.
N.B. There is the potential to create havoc in the OS that may make a system completely unusable. Also, check any downloaded utility at VirusTotal.com.