The fact that you are still able to read SMART is actually a good sign as far as professional recovery is concerned. If the data has value to you (worth $400-600) then it's worthwhile to send out for professional recovery. It may not be that expensive at this point and has a very good chance to succeed. But, be aware that the more you mess around trying things, the lower you chance of ever getting the files back will be. DIY should only be attempted if you're certain that you're not willing to pay for professional service.
That having been said, if you are intent on DIY, Linux isn't a bad idea. But, you shouldn't just try to mount the drive and copy files. I know a lot of guys will probably try to argue this, but they also haven't fixed thousands of broken drives to recover data like I have. Your first order of business should be to clone the entire drive to another healthy drive of the same or larger capacity. To do this, you should use either ddrescue (
guide here) or you might try
hddsuperclone. Both run in Linux and both are specifically designed to clone drives with bad sectors like yours. Don't attempt to clone in Windows under any circumstance! Windows isn't well designed to handle drives with bad sectors.
The reason that this is better than just trying to mount and copy/paste has a lot to do with HDD mechanics. Reading sectors sequentially (from 0 on up) is far less straining to a drive than thrashing the heads around trying to copy a million files all stored in different places. Plus, ddrescue and hddsuperclone are able to detect bad sectors and will intelligently jump over bad areas on the first pass. This prevents the drive from continually reading a bad area over and over until it kills the heads. Then, after the first pass, these programs will come back around and try to re-read within the damaged areas to get as much as possible.
After you get a good clone of the drive, you'll want to run data recovery software against the clone. It's unlikely it'll mount, and you definitely don't want to do something stupid like running chkdsk against a drive you need data from. But, any decent data recovery software will likely be able to find all your data afterward.
If you want to go the professional recovery route, just let me know where you are located and I can probably recommend a good company near you. I correspond with other data recovery labs around the world on a daily basis.
Also, just a note about your specific model. This model I believe is a terminal locked drive. Seagate basically password protected their latest models so that only their data recovery company can work on them (can you say antitrust?) and modify their firmware code. It makes them nearly impossible for data recovery labs to work on as there are only a few companies who have managed to crack this protection. Right now, recovery should still be possible w/o needing to use the terminal. But if it's pushed too hard and the heads die, recovery will be near impossible and very expensive. Thus the reason you should weigh carefully if you want professional service now, before it is too late.