When you migrate to the new Google Sites, cabinets will be converted to Google Drive folders. That is the only option available within Sites.
There is a lot of discussion on the Google forums about cabinet migration. According to this thread you should get the following warning when trying to migrate:
File cabinets detected.
Your site contains file cabinets. Files stored in file cabinets will be added to a new Google Drive folder. The new folder will be named after the converted site and will be embedded in the new draft. Links in the file cabinets will be placed on the page above the embedded folder.
So in other words once you migrate, the files that were in your cabinet will appear in your Google Drive, and you will need to manage them from within Google Drive from then on.
Another thread offers confirmation that cabinets are gone now:
The new version of Google Sites does not offer anything like file cabinets, but you can embed Google Drive folders.
Gleaning from a post on a Stanford group, you will need to manually set your Drive files to be publicly shared after you migrate:
August 13th, I received an email from someone I didn't know, asking me to grant access to one of my attachments. That's when I discovered NONE of my attachment files were accessible by the Public. I then discovered there wasn't an easy way to grant read-only access to everyone. My attachments were in Google Drive, and only I had access. Furthermore, File Cabinets do not exist in new Google Sites.
There was a person who lost 300 in-text links as they converted their site to New Sites, so it's clear that when you migrate, you lose in-text links too:
About 6 weeks ago I did a test conversion of my site "ocarina stuff," which has a file cabinet. When this page converted, 300+ links (all to file cabinet files) disappeared. [...] I really don't want to re-create 300 links from scratch.
So to answer your questions:
Will links to those files still work after converting the website to the new version of Google sites?
No, those links will definitely be broken.
Will it be possible to edit/add/remove files in the converted file cabinet?
Your cabinet will become a Google Drive folder. You will be able to edit/add/remove files from that Google Drive folder, and you will be able to embed the Google Drive folder on your pages. However, you will no longer be able to provide clean HTTP links to those files - you can only provide links to Google Drive.