Can I save my skyrim folder from steamapps to an external harddrive and transfer the data to a new computer to avoid downloading the game again?

Most likely, In fact I usually back up games I mod by copying their steam folder after a fresh install. I know steam has a backup feature, I just don't like it.

Yes, you can copy the entire folder over as indicated. When you log on it will replace missing registry settings and probably have to install its own flavor of DirectX but that's all. Very little downloading.

Steam has a backup system. Follow these instruction https://support.steampowered.com/kb_article.php?ref=8794-YPHV-2033 then run the steambackup application in the backup folder.