Could I install the 1.6.4 mod and then run that version of Minecraft? Do I need Minecraft Forge to run this type of mod?
Yes; if you want to use a 1.6.4 mod, you'll have to use 1.6.4. If I'm looking at the correct mod, it doesn't look like it uses Forge. Follow the mod's installation instructions.
Keep in mind that the mod you're talking about does seem very similar to Morph, though, which is available for 1.7.10 (and uses Forge, so it's much easier to install). You might want to take a look at Morph if you haven't already.