If storage is a problem, then don't copy them all at once to your Mac. Move a few songs, sync to iTunes match, then AFTER they've been sync'd delete them from your hard drive only. Repeat as needed.
It's a hassle yes but based on many discussions that have been previously posted here, I'm under the moressions that iTunes match works better if you don't give it too much work to do at once. That is, if you have a really huge library, it am take much more time to match your music and if any one file can't be uploaded it'll stall the procedure for a potentially very long while.
This topic has been covered in-depth on this forum.
With iTunes Match enabled the location of the iTunes library does not magically change to being in the cloud. Your local library is untouched so iTunes is still looking for the local music files to play, which it can't find because the external HDD has been disconnected. If you wish to run with the external HDD disconnected and only stream from the cloud then you'll need to create a new, empty iTunes library by launching iTunes while holding down the Option key. Choose new library. Enable iTunes Match on the new library, let it do it thing. Once it is finished you will be able to listen to the music from the cloud.
As to the trouble with the Apple TV, I suspect the problem is that you are trying to play music via the Computers menu (Home Sharing) and with the external HDD disconnected the songs are not available. With the external HDD disconnected you'll need to use the Music menu to play your music from the cloud on Apple TV, or once you follow the above steps it will be streaming from iTunes, although that will be kind of redundant in my opinion.