My photos are kept on an external hard drive, which I do not have on me at the moment. If I delete photos from the Lightroom Catalogue and move photos from one folder to another within the Lightroom Catalogue will all my deletions and movements update on the hard drive (which has my photos on it) when I plug it in later?
No, that won't work. The catalog is just a database that keeps track of where images are located and the adjustments that you have made to those images. If you delete the photos from the catalog then Lightroom no longer is aware of them and will not have any reason to try to keep track of them. I don't think you can move images without the external hard drive connected. What you will get is badges on the images indicating that they cannot be found and you will be able to move them or make any adjustments to them.
If you have created smart previews on your computer that you can add it when the external hard drive is disconnected then any adjustments that you make will be applied when the hard drive is connected again. But if you delete or move images while that drive is not connected then you will have to spend a little time getting things reconnected. It can be done but I wouldn't recommend it.