The "normal" way to sync an Android to a Mac is to embrace the cloud and do everything through Google.
If that's too 21st Century for you, then you have some other choices, they range from bad to worse:
1) "The Missing Sync" from Mark/Space - it'll cost you $30-$40, and you'll be using lots of swear words while setting it up, but it mostly works. It can have trouble with recurring events in the Mac calendar and their tech support is weak, but it's the best game in town. It's the only one I've read about that is mostly reliable and mostly supports bringing data you added on the phone back into the Mac.
2) "Syncmate for Android" - it's in the play store, but after I read the user reviews I decided it wasn't worth the trouble.
3) That's it. There are apps that sync iTunes music and a few other things but nothing else for contacts/calendar.
As for file sharing via USB - check the Android settings for "USB Mode" - there's Charge, Sync Media mode, and USB Mass Storage mode. You should be in mass storage mode, and then the Android should appear as a disk to the Mac. Try different cables or different USB ports on the Mac. Watch our for USB cables with only two wires, those only work for charging. Try with or without a USB hub if you have one. Try someone else's computer. Try unplugging the mouse (don't laugh - I've seen buggy mice that disrupted the USB bus and prevented external disk access).