Yes spider is related to spindle but it is way less obvious, since the English language dropped the -n- and kept the -d- when referring to the animal. But dropped the -d- and and kept the -n- when referring to the activity. So you end up with the tool: “spindle”, the animal: “spider” and the activity “spinning”. Most Germanic languages ( I’m going to use German as an example) dropped the -d- when referring to both the animal and the activity: Spindel, Spinne und Spinnen.
Ah, nice!