1. NTFS is typically a read-only setup. There are projects and kernel options to allow it read-write but they are specific, some require exactly the same file size be kept (you can change a "u" to a "t" but cannot remove the "u" without adding a byte), others have other requirements, but generally speaking, most Linux users setup a Fat32 filesystem to do cross-OS file transfers/editing. It allows read and write access to the files on that filesystem from both operating systems.
2. Your last one: cd Desktop/ should work, assuming Desktop is a directory you can enter. Try this as a good check:
This will create a directory with the correct permissions (hopefully) and allow you to enter that directory. If you can enter it, then you have determined it is possible. Next step is to find out why you can't enter "Desktop". ls -l will display the permissions of the files and directories in the current directory. You need the execution bit (xwrxwrxwr, x is the execution bit) for your current user or group (the first and second xwr xwr instances) to have access to any given directory. If you have access, then your command may be wrong, look again and make sure the directory is Desktop and Dsktop or something else.
3. Sounds like a hardware setup issue, but I'm sure you've tested it from windows. All I can say is check alsamixer again, make sure it's not muted and that you have both PCM and Master volumes turned up.