Current versions of fileMaker use a different file format. In fact, from V3 there have been about 5 file format changes. The current version of FileMaker, Version 12 cannot even open this file nor convert it to its format.
FileMaker 11, however, can convert FileMaker 3 into a .fp7 file--the format used for FileMaker versions 7 through 11 and it can be converted into version 12. Thus, you may be able to either borrow a copy from somwhere or get FileMaker customer support to make available a 30 day trial copy of FileMaker 11 that will enable you to bridge the gap between 3 and 12. (Or since 12 is brand new and has a numer of unresolved issues, you may want to buy FileMaker 11 and convert to it instead.
FileMaker Pro still supports peer to peer networking for up to 9 client machines. Mac should open the converted files and be able to run them about as well as a PC could but there have been a great many changes to the DB so you will need to test the converted file extensively to resolve any issues created by the conversion. You may even be better off exporting the data from the client's current copy into a tab, csv or merge file and then import the data into a brand new file designed to your client's needs from the ground up in either FileMaker 11 or 12.
If you are going to use Peer to Peer networking (One copy of FIleMaker opens and hosts the files for the others), the closing the file and using a third party utility to back up is still a good option. You can also write a script that uses Save a Copy as to save copies of the database file to a designated location--including a hard drive on a different computer on the network. If you upgrade to FileMaker server, a schedule in server can automatically save backups on a regular schedule that can be daily or even hourly.
Backing up information using just FileMaker Pro is not a feature. You would need third party back-up software in order to accomplish (You mentioned Mozy, Carbonite is another option). Also remember, best practice with third party back-up software to to close the database first.
Also, please note that FileMaker Server products can schedule backups as a built-in feature. Just something to think about.
Backing up information using just FileMaker Pro is not a feature.
But is something that can easily be scripted to take place on a regular schedule or via a script that runs when the last user closes the file for the day.
Here's one example script: Saving Sequential Back Ups During Development
The 15 minute interval is for a file undergoing development. For a production database, you'd use a much longer interval--up to 24 hours and you'd only run this script on the host machine.