Full wave rectify the supply to all the lamps and switches so they are running on unsmoothed DC so you can use diodes to implement the two 84 input, and single three input OR gates required. Use 24V DC coil relays, and if they chatter, add just enough capacitance across each coil to hold up the coil voltage through the zero crossings. Don't use too much capacitance, as in the limit, it will increase the average coil voltage by a factor of up to 1.4 and cook the relays!
As Benta has pointed out, section 4 wont work as drawn. Either swap the thee NC switches in series to three NO switches in parallel, with the whole switch group in series with the relay 4 coil, or put the coil in series with the NC switches, and wire the contacts based on its normal state being energised and releasing on any button press, or use an extra relay to energise relay 4 when it releases, or (assumiong DC supply) use a transistor, a couple of resistors and a diode to implement an inverting driver for the relay 4 coil controlled by the three NC switches.
N.B this all assumes you've either got access to *BOTH* ends of the transformer secondary supplying the 24V AC, or can rewire the ground side of all the lamps to the negative terminal of the bridge rectifier. If not, so all you've got is ground-referenced AC, it gets a *LOT* harder . . .