Have you checked permissions on the server? Under System Information in Joomla, you can check the permissions for common folders. If most appear in red color, then there is probably an ownership issue for files and folders and your hosting company should correct it.

OK, first things first. When you use the "Image Gallery" tab in K2, all the uploaded images are moved into /media/k2/galleries/ (in subfolders based on each item's ID). The reason this is done is similar to how images upload via the "Image" tab are moved into /media/k2/items/: keep the media out of the users reach so no accidents happen (e.g. deleted media). This kills re-usability, but experience has shown that content managers usually upload new images and most times keep a pool of media/images under /images in Joomla, if they want to re-use something.

Now, if you create "regular" image galleries using the "Simple Image Gallery Pro" editor button, then these image galleries will be stored under the "root path" that you've defined in SIGPro's component settings (the default is /images/).

I hope you see the distinction already between the 2 ways of inserting an image gallery in K2.

As for the errors you get, these are JS related and they could be caused by other plugins that can load in K2 pages (e.g. system or third-party K2 plugins).

Can you share 2 screenshots please? One with the form after you save the item and one with the browser dev tools in a separate window, again after the item is saved.