I am considering buying a new Canon EOS M3 to use my very old Canon FD lenses. Would they fit?
They won't fit directly on the mount, but you can find adapters for the Canon FD mount for all the mirrorless camera mounts, including EOS M. I'd actually suggest looking at some of the other mirrorless mounts, such as Sony NEX, Fuji X, or micro four-thirds, considering that the EOS M2/M3 and the EOS M ultrawide zoom are only sold in Asia, so the upgrade path, and the native autofocusing lens pool are much smaller than with other mounts.
You will also want to consider that the EOS M has a crop factor of 1.6, so the lenses will behave a little differently than they did on your Canon FD bodies (e.g., a 28mm lens isn't very wide any more on APS-C).
In addition, because the FD lenses are fully mechanical, you won't get any electronic communication between the lens and the camera. No autofocus. No shooting in modes other than M and Av. No EXIF information from the lens. Consider also that the EOS M has no focus peaking, only magnification (although, if you're brave enough to go with Magic Lantern, that could help), and there's no viewfinder.
While you can use FD lenses, it's much more of a pain than if you use a native autofocusing lens in the mount, and it will be quite a different experience to using them vs. shooting with them on a Canon FD-mount SLR.
Not directly, no.
But you can buy an adapter that lets you use FD lenses. The adapters are rather cheap and available in most online shops. (amazon, ebay, ...)
I don't own one so I don't feel like linking to a particular product. Search for "EOS-M to FD adapter".