If I (or a friend, or more likely, a 555 timer) press the capture button for a few seconds every 15 seconds it will create a bunch of 15 second clips. If I then splice these together, I get one continuous video. Is this allowed? (I don't want to spend $30 USD on a capture card).

This is not allowed, it must be 1 continuous video

If you don’t want to buy a cap card, pointing your phone at the tv and recording that way also works