Designing a safety circuit to ignite a pyrotechnic device
All of these three detection methods can be done with some simple CMOS logic circuits, take a look at flip-flop or set/reset latches as a good place to start.

Each of the three inputs could be separated into individual tests, then be latched and the outputs of the three latches ANDed together, so the ignition would only occur once all three signals are present.

The moment all three are present (and therefore each of the three individual conditions has been met), the AND gate can be used to both trigger the ignition and reset all the latches and return the system to it's standby state ready for the next activation.

Of course, all of this is theoretical and safety needs to be taken into account. In industry, this is usually done with a combination of things, most importantly a dead man's handle (as already described) and using the concept of a second channel of independent control. Often this is done with a software channel (doing all the tests in software, perhaps on a PIC), mixed with a second hardware channel completing the same tests using low-level logic (perhaps CMOS or relay interlocks).

The overall safety of the system is usually designed to meet a particular Performance Level (PLd) as defined in the EU by the Machinery Directive.
Depending on your intentions with the project you will need to comply with a variety of different regulations and directives so of course all of this advice is given with that in mind.