WARNING:
Furnaces are expensive, complex pieces of equipment. If you don't have the proper tools and/or knowledge, it's often best to let the professionals handle maintenance and repair.

Gas Furnace Ignition Sequence:

1. Thermostat calls for heat.
2. Pilot gas valve opens.
3. Ignition control starts (spark or glow).
4. Pilot gas ignites.
5. Flame sensor detects pilot flame.
6. Main gas valve opens.
7. Main burners ignite.
8. Blower timer starts.
9. Flame sensor detects main burner flame.
10. Blower starts.

High Efficiency Gas Furnace Ignition Sequence:

1. Thermostat calls for heat.
2. Draft inducer starts.
3. Vacuum switch detects negative pressure.
4. Pilot valve opens.
5. Ignition control starts (spark or glow).
6. Pilot gas ignites.
7. Flame sensor detects pilot flame.
8. Main gas valve opens.
9. Main burners ignite.
10. Blower timer starts.
11. Flame sensor detects main burner flame.
12. Blower starts.

Ignition Problems:

Vacuum Switch Not Closing

In a high efficiency furnace, if the vacuum switch does not close after a certain timeout. The furnace will shut down, then possibly retry a few times depending on model. If after a given number of tries the switch still does not close, the furnace will enter lock out. Once in lock out, the furnace has to be manually reset (depending on model).

The vacuum switch will look something like this...

And will have a rubber tube connecting it to the draft inducer. This part cannot be repaired, and must be replaced if it's faulty.

Pilot Not Igniting

If the pilot does not light, you'll first want to check to make sure there is something trying to ignite it. Typically a spark or glow ignitor is used, so first you'll want to determine which is being used.

Glow Ignitor

A glow ignitor looks similar to this, though may not be visible without further disassembly.

It works by passing a current through it, causing it to heat up and glow. It should heat up enough to ignite the pilot gas. If it doesn't heat up, it should be replaced.

Spark Ignitor

A spark ignitor looks like this.

Take notice of the thick, often orange or red wire that is typically used to connect the ignitor to the ignition control module. This device works by generating an electrical arc between the two probes, causing the pilot gas to ignite. If there is no spark, you'll have to replace the ignition control module and/or the ignitor itself. If the ignitor is not igniting the gas, but there is a spark. You can try cleaning the electrodes with fine steel wool, to remove any carbon buildup.

Pilot Not Proving

For a pilot to "prove", it simply means that the flame sensor has sensed the pilot flame. If there is no pilot or the sensor doesn't detect it, the furnace will often purge the system and then try again. Furnaces will often try a set number of times, before entering lock out.

If you can see the pilot flame, but the sensor is not detecting it. Try gently cleaning the sensor with fine steel wool, to remove any built up carbon. Also make sure the sensor is in the proper location with respect to the flame (Check the owners manual for proper placement), and adjust as necessary. If that doesn't work, replace the sensor.

Main Burner Not Proving

If the main burner ignites, but the furnace shuts down before the blower starts. You'll want to check the main flame (or rollout, high limit, etc) sensor(s). If your furnace has error code indicators, check those and compare to the owners manual for translation. In this case, you'll want to test and/or replace each sensor and/or contact an HVAC technician

The answer will depend on how much temperature drop/rise there is inside the ducts. A two stage cooling system means that the heatpump will either drop the air return temp a little bit (stage 1 is running, half cooling) or it will drop the air temp a lot (stage two is running, full cooling). Either way the fan moves the same amount of air. In stage 1 mode the net effect will be for it to spend more time with the fan on, since to hold say 70F it will have to spend a while pushing 65F air through, vs a single stage system that puts out 60F air which will get back to setpoint faster. More air circulation (assuming its all a nice 65F) should actually be better for comfort in your restricted upstairs area vs spending a short time blowing 60F air until the downstairs is comfortable, leaving the upstairs without much time to get cooled air.

If the ducts are soaking a lot of the cooling up (i.e. the downstairs is getting 65F air but the upstairs is getting 68F air) then no, the first stage of a two stage unit will not give a lot of comfort and a system pushing colder overall air (i.e. single stage) all the time would be preferable.

In any case (if you get this system or not) most ducts, even in older homes, have some sort of internal damper to control flow to each run/room. These might not be accessible if you finished your basement and covered them all up, but that's for a different question about exposing louvers and patching sheetrock ;-). Further, your vents in each room should have a shutoff (not preferable vs in-duct damper, because it's noisy) which would allow you reduce the airflow to the downstairs rooms. This would be the solution if you find that running the system just doesn't get the upstairs rooms comfortable. Start by reducing flow to the one or two ducts closes to the thermostat, and continue until your system runs long enough to cool the upstairs without leaving any of the first floor rooms too frigid.

edit: grammar and clarification