Some general beginner questions that I have been saving up..

I expect others will show up to express their opinions and hopefully also help expand upon my answers, as I'm sure I'll miss a few points, but here's some thoughts.

• What is the difference in "function/purpose" between the first rise and the second rise? 
• To clarify, when I hear the phrase "first rise" or "second rise", my definition of first rise is "bulk fermentation", or any rising/proofing/fermentation that occurs prior to dividing or shaping, while I take second rise to refer to the rising that occurs after shaping. Briefly, I think the bulk fermentation allows the yeast to establish itself in the dough and generate gas, allows further hydration of the flour, and relaxing of the gluten after mixing. The second rise is mostly about CO2 production so that your bread comes out of the oven with the desired lightness.
• Is the second rise called "proofing"?
• Yup. Many people call it that, though proofing can also refer to the first rise. I use rising/fermenting/proofing pretty much interchangeably when talking about bread. I think some get more specific about which term they use when, but they're all valid ways of communicating what's going on with your dough.
• What are the symptoms of under (1st) rise? and over (1st) rise considering the second rise is timed correctly?
• Well, unless you're giving no time at all between mixing and shaping (and thusly not allowing the dough time to relax), the main symptom of underdoing it on the first rise is that they'll be less gas in the dough, and you may need to give it a longer proofing after shaping to compensate. I've seen recipes (some bagels, some rolls) that actually do call for that, though rare. A few minutes (10 - 20) is usually helpful to make the dough less resistant to being worked with in these cases. Generally, it's better to underproof a little than overproof. In the case of the first rise, overproofing at this stage may mean that the yeast consumes too much of it's food and doesn't have enough energy to have a good second rise, and may ultimately come out denser than usual.
• As far as simple sandwich bread is concerned, are there any advantages or disadvantages with increasing the hydration? (sometimes when I go by the recipe, the dough seems stiff to me, and yes I use a scale)
• I think sandwich bread recipes tend to have a lower hydration for two reasons. 1) They're often enriched, and those enrichments usually bring extra hydration with them (eggs, butter) or fats that can hinder gluten development (oil, butter). 2) Drier doughs tend to have a finer, more even crumb which is better for spreading things on or putting fillings on, without stuff falling through holes. You totally can take a recipe for a wetter dough and put it in a loaf pan if you want; people do it all the time. If you're making a recipe you've made before and it seems drier/stiffer than usual, feel free to increase the hydration slightly. If it's a less familiar recipe, well, it may be what the author intended in order to get their desired result.
• When preparing the dough for the second rise/shaping, how gentle must I be? I mean do I perform a second knead or just gently flatten out the air bubbles and then shape?
• You don't need to knead it again. Just flattening out and degassing it a bit is generally the way to go. How much you do that depends on the bread. For sandwich loaves, I usually go fairly flat and try to degas a lot. Sometimes I'll even take a rolling pin to them. For hearth loaves (such as baguette), I usually try to get it flat enough that I feel like I can shape it, and get rid of some of the really big bubbles, which will only get bigger later. Actually, I think I degas baguettes some what moderately; I'm not being as diligent as for a sandwich loaf, nor as careful and gentle as I would be for an airy boule or some ciabatta.

Hopefully that was fairly clear and helpful. Feel free to ask more questions if something was unclear or piqued your interest.

I am a total dunce. Do I measure the flour by the grams, which is 128 grams a cup? or 4.5 ounces on my scale. Anyway, this is what I did this time for the first recipe. 3 cups flour would be 13.5 ounces of flour on my scale. I know nothing about baking and had someone talk to me about correct measurements. If I had seen the 3 cups of flour before, I would have multiplied 3 cups x 8 = 24 ounces, and I would have put that much in. I don't think that is what you had in mind for the first recipe, right?