1. Some apps can't be moved to the SD card. ("Moved" is a misnomer - small parts of apps are actually moved. Pointers to those parts are left in internal storage. So unless a part is considerably larger than a pointer, you haven't really "saved" any space.)
2. The card is meant for saving files on it. Music, pics, vids, etc. Use any file manager to copy the file(s), paste them into a folder on the card and, if the copy is successful, delete the original from the phone. (A move that goes bad leaves you with no file, a copy that goes bad still leaves you with the original, so you can try again.)
3. I'd personally move all the files to a folder on my PC, remove the card from the phone and connect it (just the card and whatever adapters you need - no phone) to the PC, then copy the files to the card. Some versions of Android won't let anything (including a connected PC) write to a folder unless that thing created the folder or the folder gives the user 'everyone' write permissions. The PC doesn't know about such things, so it just writes the files to the card. Put the card back in the phone, turn the phone on and it finds the new files.