Deleting Old Library folders

Understood. Here's what I'd do in your situation.

1. Get a search app for Mac: I use easyfind, but I'm sure others are available. It finds things that Spotlight doesn't
2. the app can be configured to find files by name, suffix, contents, etc. Spend a bit of time understanding how to set it up - wildcards, directory search, etc.
3. Use the app to search for the sort of files you're concerned about. Search for .doc, .xls, ,jpg, .tiff, ... etc. It will take a bit of time, but it will find all the docs, pics, music files that are there and not embedded in Apple libraries with odd formats or naming standards. With a bit of googling you might also be able to identify files with Apple-specific suffixes - like mail messages and attachments and such - but I've never tried.
4. You can sort the search results by name, size, type, etc. which can be a quick way to find duplicates (which you can delete) or files you don't care about (plists and other config files) which you can also delete
5. The app will also search for content inside files (takes even longer) so if you've got stuff you remember but can't find you can search based on content that might be there e.g. kids' names, pets, and so on.

It's a bit time consuming, but you can leave it running once you get used to how it works.

Note that this might not find things that Apple seems to store deep in its libraries with its own naming formats - like books.

Good luck.

I wouldn't, if I were you unless you are really certain what's in them or what they refer to. I appreciate that you've come here to ask that, but it's going to be hard to get a definitive answer depending on what apps and data you use. Podcasts and Books are stored in the library, as are Kindle books. In the old days attachments to mails were stored in a separate folder from the mail messages; I think that might have changed, but be careful.

If you're doing this to save space then you could start by sorting by folder size and trying to work out what's in the larger ones; deleting loads of small settings folders won't save significant space and you risk messing up apps that you've worked with for years. The other thing is to find library entries for apps you've deleted and get rid of their traces, but it's a pain cos many of them are locked and need login or terminal and sudo - and 90% of them will be <1MB, so hardly worth going after if space is your issue. You'd need to use a search app to find them because Spotlight doesn't find them.

I'm a bit OCD about keeping my disc tidy, but I don't mess about with the library.

I can't really advise specifically, but if they are really backups then does it matter. Being pedantically for a moment, backups are copies of live or recently live information for apps and data that is currently in use. If your Mac died tomorrow you'd use your backups to recover it to a state almost identical to before the crash. Archives are old data that's not in use any more, but is being kept for sentimental, legal, fiscal reasons and so on.

Where are the folders in the screenshot? If they are in a time-machine backup then I don't think you should delete them, although I'm not a time-machine expert. If they are just a bunch of folders backed up somewhere then I don't think you've got much choice other than to go through them one-by-one and trying to work out what's in them.

If it were me I'd just buy a disc, dump them all on it and store them somewhere safe. Rule number one about backups/archives is that you only discover you really needed a file about 24 hours after you deleted it.

Spotlight doesn't always find hidden files. I don't know if it finds them if you show.