Macbook Air Back up Question

I've done this myself. Just create 2 partitions using Disk Utility and use one for Time Machine.

 

Although you shouldn't use a backup disk as also a primary storage disk for those files you want to share between your Mac and PC, there's nothing stopping you.

The best thing to do would probably be to partition this drive into two partitions (use Disk Utility). Make the TM partition at least 2X - 3X the size of your MBA hard drive, and the other one can be the rest. Then, format the TM partition to HFS+, and the other one Fat32. The FAT32 partition will allow you to both read and write to it from both your MBA and your PC. There is a 4GB file size limitation with FAT32, so if you are transferring video around this may not be the best.

Once you do this, connect the drive to your MBA and start TM. TM should ask you if you wanted to back up to this drive & partition; if it doesn't, launch it and set it up to use this drive.

 

exFAT does indeed work for reading/writing between both OSX and Windows, however Time Machine does not work with exFAT (only with Mac OS Extended Journaled).

As for the other issue, I think you have to erase the entire disc first before you get the option to create partitions. It's been a while since I partitioned my 1TB external (I did the exact same thing you're doing now).