No need to run away when a problem occurs. If you don't solve this issue it will return in Ubuntu.
I just downloaded lzRichEdit and tried to verify the issue, but it is no problem to modify an inc file and save it.
When the error message appears the next time could you please click into the message dialog, press CTRL+C to copy the complete error message to the clipboard and paste it here into a forum post? I am asking for this because your screenshot does not show the entire file path.
Go to "Tools" > "Options" > "Environment" > "Backup". For me the settings are "Project files" > "Same name (in subdirectory)" and "backup" subdirectory, as well as "Other files" > "User-defined extension (.pp.xxx)" > "bak" User defined extension, "backup" subdirectory. Since these are the default settings you probably have the same. One possibility could be (although I don't believe it) that the backup directory cannot be created. Does it exist? can you create it manually? Do you have write permission in the folder into which you extracted the lzRichEdit package?
Do not install Lazarus or any third-party packages into "c:\program files". You will have lots of access right issues here. Better to install Lazarus into something like c:\Lazarus (not 100% sure, but if you are an "ordinary" user you may have access here too). D:\Lazarus (on a different drive) definetely works, as well as, of course, your user folder (c:\user\<your name>\Lazarus, or similar)