Power Latching Relay off/on in response to 12V ignition and 1 second ground pulse
Here is a circuit that may meet your requirement.
Inverters are a package of hex Schmitt trigger inverters.
MUST be Schmitt trigger.
There are two spare inverters which can be used in various ways if desired.

You could convert this to a '555 based circuit with more components and less "properness". eg here the 1 second delays are driving a Schmitt triggered CMOS input. 

D1 accepts 12V positive on ignition
D2 accepts a 12V positive going alarm pulse. As the contact is floating this should not be a problem. If desired you can invert either input sense with a spare inverter. 

When either input is high, Q1 inverts this to low and D3 keeps C2 discharged.
An alarm pulse occurring during a R2C2 timeout resets the timeout.

When neither input is high R2 charges C2. A 1 minute delay could use eg about 10 uF / 4M7 . Values will need to be "select on test"ed. 

R1C1 and R3C3 plus following inverters provide 1 second delays.
A negative going edge pulls the cap low and the resistor then charges it high.
When the alarm pulse activates C1R1 the output pulse lasts the lesser of the alarm pulse or C1Ra timeout. As the set/reset relay only needs a 10 ms pulse 'this is not a problem". 







simulate this circuit &ndash; Schematic created using CircuitLab

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Added:

Extra inputs eg floating/ground on/off

If you follow my circuit and see how it works it should be possible to add to it with the spare inverters or modify it. 

"Floating" inputs are best pulled to non active state with a pull up/down. This can be behind an input diode (see eg existing Ignition-D1-R6 where D6 pulls the input to ground if Ignition is open) to ensure input idle state is defined. 

If voltages spike too high, clamp them to relevant rail (12V or ground) with a diode.
eg in existing circuit the input to IC3 spikes to 2 x supply when IC4 out goes high.
A diode across R3 (Cathode/bar to V+) stops this.