If you are using a computer, you can rearrange tracks in a playlist by going to the playlist’s edit page. You can access the edit page at any time by clicking the pencil icon below the playlist’s banner. 

Go to the Tracks tab, and you can drag and drop the tracks into your desired order.

If you are using a mobile device, Android or iOS, tap the “edit playlist” in the settings dropdown on the playlist and drag and drop the tracks into the order you want.

Removing a track from a playlist:

On Web, go to your Tracks tab, click the small x on the right side of the track and save the changes.

If you are using an Android device, click the 3 dots and tap "Remove from playlist". On iOS, swipe left and tap "Remove".

Rearranging playlists:

There isn't an option to rearrange the order in which your playlists are shown on your profile. 
The playlists on your profile will always be ordered chronologically.