Or can I just spam them out as fast as my program can send them, as, like you said, you cache it anyway.
No. The throttle discussion page specifically says:
... we consider > 30 request/sec per IP to be very abusive and thus cut the requests off very harshly.
And even though the underlying data is cached for 60 seconds, the HTML requests are not. You'll still burn one quota slot with each call, cached or not, and risk the wrath of a backoff flag, or worse.
See also: ["Is it possible to access data in real-time using the Stack Exchange API?"][2]. Once per minute is a pretty reasonable rate to repeat the same request. Multiple times per minute also risks running out of quota -- especially if you make several calls per batch/poll.