the battery specs will tell you the recommended charge current, but roughly speaking:
(time to charge in hours) = 1.2 * (capacity in Ah) / (charge current in A)
If you wanted to charge it in 24h, the current would be approximately 1.5A
As long as each mosfet is on a different heatsink (or you're just using a single mosfet) you'll be OK for testing.
They shouldn't get too warm. They should dissipate about half their rated power. Start with just a single 2R2 resistor. This should give you a 320mA limit.
Design for the...
There is mailing list dedicated to repair of old HP and Agilent hardware.
Of you're not familiar with it I'll look out up.
It was a Yahoo group from memory.
You need to know the voltage and wattage. It should be written on the old bulb.
If you have no old bulb you can (very carefully) measure the voltage on the socket and then choose a wattage that is bright enough.
After a digikey search I'd recommend a IPA50R199CP
It has an insulated tab which will make it easier to place these on the same heat sink. Otherwise you'll need insulating mounting hardware.
the diode must be reverse biased across the relay coil.
the relay should have a 12V coil
The mosfet will need to be a logic level mosfet (what mosfet have you chosen?)