Hey there! As always, please note that I don't work for Lockitron, but having been around for a while, I can probably answer a few of these questions well.
Regarding the way iOS and Location Services work, see this post I wrote not too long ago:
Regarding your specific questions:
What steps I can take to make sense work?
Make sure the Lockitron has not been force closed (ie. it shows up in the multitasking view, do not swipe up to get rid of it), and make sure that it has all the permissions it needs. As a quick check, go to Settings, scroll all the way down to Lockitron, and make sure everything is on.
You'll need at the very least:
Location - Always
Motion & Fitness - On
Background App Refresh - On
Also, is there a way to use lockitron sense without GPS? Could the sense feature use the presence of the lockitron bluetooth signal (which I assume is relatively short range) to trigger? And then reset itself anytime it loses that signal?
As mentioned in my post above, to save battery life from constant searching, it uses the geofence to know when to start seeking out your lock.
Also, would it be possible to use the lockitron API myself to hack together an app that does make it work this way? Perhaps in conjunction with a beacon inside my house or something?
You could do this, but then you're just shifting the constant searching to another app, and then you're still using battery life. Apple's own Store app uses iBeacon to do something similar, but it still has to go through Location Services
Finally, the tap on your phone to open the door feature -- which I haven't experienced myself yet -- what's the reason you can't set your phone to simply always lock and unlock your door in response to a triple tap? Do the iPhone APIs only allow a limited time window where global shortcuts like this work?
Make sure vibrate is turned on, and make sure the notifications are on for the app. Without notifications, it can't buzz you to prompt the tap to unlock. And again, as per my post earlier, yes, there is only a limited time window where this would work, so that's why there's no "tap to lock".