Given that a decent alarm system usually has the capability to send alarm notifications by phone (which tends to get replaced with "over Da Intarwebz" though), SMS should be a viable option for most of the requirements you list:
[x] * Check that I set it after I've left home.
[x] * Enable it if I find I need to later on (e.g. if I did forget).
[x] * Disable it if I need to (e.g. my partner returns when I'm not around / I want my neighbour to check on something for me / I have a delivery or service person I want to enter my home whilst I'm at work)
Limit control for this option to phone numbers registered with the control unit, and add a one-time code (copied from the control unit before you leave) if you're worried about number spoofing.
[x] * Be notified immediately on my phone that it has been triggered and take appropriate actions such as calling the police / turning the alarm off if it's a false alarm or it's done it's job and I want to stop driving the neighbours crazy / logging into cameras in the home to see what's happened)
[x/ ] * Have more than a rudimentary All or Nothing approach to my home security. (E.g. different access levels for different people / ability to amend these on the fly as needed).
Amending on the fly is explicitely something I wouldn't want. Security, including access modes and zones, is something I'd design and set up beforehand. If I then need to grant access in a way that doesn't match those predefined modes, then it's "tough shit, come back tomorrow".
I want my home control system (which an alarm can be considered part of, although not necessarily integrated with) to offer a limited number of predefined states, such as "I'll be home in half an hour, set the living room thermostat to $preset(comfortable)" unless you have direct physical access. And controlling the system from outside the house can only select the applicable subset of those predefined states.