Why would you think you would have to buy it again? The licence for windows is tied to your motherboard... and since you have windows 8, you can always unlicense your computer as it and reinstall with your key on a different computer.
That being said, DO NOT TRY TO JUST MOVE THE INSTALL. That will cause a LOT of issues. Instead, here's what you want to do. First, back up all the data (documents, save games, anything like that) to an external drive. Then unplug your hard drive, and plug in the SSD - make sure it's on a SATA III connection if you have it. Then put in the windows 8 CD, and select 'clean install.' Go through that process, then ONLY AFTER you have windows completely updated to 8.1, with all drivers and security patches, shut it down, and put your hard drive back in. Then format the hard drive and move your data back to it.
Yes, you will have to download updates again. You will lose your settings. You will lose drivers. But really, that's not a big deal; they don't take that long to set up again.
As for "How do I remove Windows 8 from my HDD without losing all my music, pictures and documents and without a backup?" ...give me a break. You use a backup. That's how that works - no other way is failsafe. If you don't want to get an external hard drive to move things to (which you should have anyways, because if your drive fails, you would lose everything on it), then you can take all the data you want to keep around very badly, compress it with a program like WinRAR, and send it to a cloud storage service like dropbox or google drive.
You will be able to install only battlefield 4 to the SSD, but you won't see much of a benefit from it. The reason you would put a game on an SSD is to get rid of loading screens; that's not an issue in BF4. The only games I would recommend putting on an SSD are MMOs and single player games.
128GB will be plenty for Windows 8 and a couple games, yes. However, that's a cruddy pick for an SSD... consider that Newegg is having a sale right now on the
Samsung 840 EVO for $90, and you should leap on it.