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I feel there has to be someone more versed in SP than myself, but having recently (<6 months) migrated from efolder to on-prem SP for a new client I also feel I know a little bit (just enough to be dangerous).
In short, if its collaboration that they seek and external access is needed I would get a demo of AutoTask workplaces up and running for them. It does 99% of what they would want except integrate into AD for a SSO thing (maybe they do, I didn't ask).
If they must absolutely go forward with SP, here are some things I learned:
File names must be formatted as if they are going to be a URL. By that I mean no longer than 128 characters and can not include certain characters (#$%& etc...)
Further reading on this (potential) issue, this is highly suggested: https://technet.microsoft.com/en-us/library/ff919564(v=office.14).aspx https://support.microsoft.com/en-us/kb/2933738
SP is designed around the idea that you access it through IE. You can access it through windows explorer (and map a drive letter to your site using https) which simplifies the interface greatly, but makes it very unusable. If the user runs into any error, they will receive a cryptic "you cant do that" or "this failed" message that is frustrating for them. The same errors in IE are handled much better and deliver usable troubleshooting information that allows the user to resolve 50% of their perceived SP issues. Plan on training users on how to use IE and SP, they must have the basics down or else you will be flooded with tickets when it launches.
If SP is replacing a file server, keep in mind SP library's are limited to 50,000 files (for performance reasons). Determine your file count now and devise a strategy to organize your files around this limitation.
If your users need need mobile access, we used "SPConnect" which is free and has built in viewers for all supported file types. This works on Android and Apple devices, no app for Windows phone Tips hat to developers ... Its been the only thing that has simply worked.
https://play.google.com/store/apps/details?id=com.acquirednotions.spconnect3&hl=en
We had to move about 45,000 files into SP from a efolder. efolder (by default?) caches a local version of all files you have access to, so logging in as the efolder admin gave us local access to all files in their share. We robo copied these files and verified after the move to a local folder. From their we had to scrub the files to find and replace illegal characters using powershell (https://blogs.technet.microsoft.com/heyscriptingguy/2013/11/22/use-powershell-to-rename-files-in-bulk/). This was technically easy, the hard part was getting the customer to agree to renaming all of their files since they had a very specific model for file names.
Once the files were moved (and integrity verified) and scrubbed of illegal characters we mapped a drive to the local file share ("X" drive) and mapped a drive to sharepoint using https then used robo copy to move the files again into SP. This part took over night, about 10 hours total.
Once the data was moved into SP we had another hurdle to overcome. Because we have "require check out" (Library settings > Version settings > Require check out) enabled to keep users from having "collisions" with files, ALL of the files moved into SP were required to have an initial check-in before other users were able to see the files. To achieve this bulk/automatic check in of 45,000 files we relied a sharepoint workflow that basically checked the "check-out" status, has never been checked in, if this was true it would put a "space" in the meta data field and check it in for the first time.
Once that was done, the rest was a matter of verifying data/files, assigning permissions and testing with pre-selected users before going live. If you have specific questions, fire away.
It was a huge hassle, we all agreed now that we would never do it again and if we need something like this we will use AutoTask Workplaces. AT workplaces gives MRR, WAY easier to setup and has better features/easier management.