Will the current extensions and my code risk being broken? Should I consider backing up my current version of Inform 7?
The new 6M62 version looks quite good, though. 300 bug fixes! And changelog in Epub! 
You should be able to redownload old versions of Inform from the website if your code turns out incompatible with the latest release. It’s always good to have a backup, but in this case you’d want to be backing up your code/project-files, not the software.
There were a lot of changes in 6L02, so if your project is older than that you’d have to rewrite quite a bit of it (but still shouldn’t take long to do, just following the error messages for a while).
You can read the changelogs for the other two versions to see what language changes they had, so you’d know if you were affected by them.
Extensions older than 6L02 mostly don’t work with 6L02 and up, too, although the public library has a lot of them updated. If you depend on some complex extensions which weren’t updated, you might not get enough from updating.
Generally we talk about “big updates” and “bug-fix updates”. 6L02 was the last big update; as reddo says, it required changes to many (not all extensions) and a lot of game code.
6L38 and 6M62 have been bug-fix updates. Most game code will come through those okay, although you might have to change wording in a few places.
Sounds good - Looking forward to the Win version!