is there any way to use the USB with some sort of adapter in order to, over LAN, be able to both monitoring and, perhaps edit, upload/download to the Arduino?
No. The Arduino establishes a serial connection to the PC it is attached to.
That PC could be running server software (Apache, for instance) and could serve up web pages to display data from the server that it gets from the attached Arduino. The server could also send commands to the Arduino.
As for uploading new sketches to the Arduino from a remote computer, that is remotely (no pun intended) possible. There needs to be an app on the PC that knows how to get the code, how to invoke the compiler, linker and uploader, and how to send error messages back to the remote computer. Possible, but not trivial.