Hi!
heres how i understand it (explained via examples):
the tax slider controls your income only - so say you have 100 income from tax. this income is entirely independent from how much you spend, which is controlled by the spending slider. The maximum you can potentially spend is defined by the number of factories and research buildings on your planets (and civ abilities).
imagine just one planet, factories + initial colony total 50 manufacturing points say. if you had spending on 50%, sliders at 100% military and 0% for the others, you would spend 25bc a turn, and it would all go to building a ship on that planet.
change the spending to 100%, you spend 50bc a turn - regardless of the tax rate.
if you then built another factory to take the total to 60 mp on that world, you would then be spending 60bc per turn building ships - which answers the second question - you would build the extra factory to give you the potential to spend more per turn. of course if your taxes (+trade etc) dont provide enough to support this extra spending, it is pointless to build the extra factory. so its all about a balance.
change sliders to 50% social, 50% military, and you will still spend 50bc - 25bc on ships, 25 bc on improvements this time though.
same applies to research -say you have research buildings giving 20tp
100% on research, and you only spend 20bc a turn. 50% mil, 50% research, and you spend 10 on research, 25 on ships so 35bc total.
once you add in civ abilities, it gets more complicated - basically same as above but add the relevant percentage to the number of hammers/beakers you produce, and only charge for half the extra.
to sum up: factorys and labs define potential social, mil and research production, soc/mil/res sliders how much of each production potential can be used, and spending slider how much of your potential spending you actually spend.
sorry if this makes no sense - at if any of its wrong i'm sure you other guys will correct me