1) If you settle 1 tile away from a 3 hammer spot you will get the bonus production if you have your citizens work that tile (you can adjust what tile you citizens work from city screen). It's generally not a good idea to work the 3 hammer tile untill your city has a reasonably sized pop.
2) The blue circles are recommended spots but the computer will usually only calculate for food (at least in my game). The computer will not take into account the tiles distance from other cits, the resources hear the spot, or whether or not the spot is located in a good political are (i.e. it might be to close to your neighbor or not situated in the position where you will obtain the most land. It's best to ignore the blue circles
3) Mountains do not affect your city incept that for the most part they are useless terrain. (In the R&F scenario/mod I think some civs can produce food on mountains).
NOTE: Going back to 1) It's usually a bad idea to settle on a 3 hammer tile that has resources, in fact the only reason for settling on a 3 hammer tile is a) proximity to resources, b) it's a good defensive spot. You will get a better production bonuses from a tile if you build a mine there which you cannot do if there's a city there. If there's iron or bronze on the tile you can even 5+ hammers (represented by an anvil) which greatly improves production.
(1) Actually, in the last few games I have got several cities with mostly hills around them which I mine. Combined with coast and a Lighthouse they can grow a bit more but they are then fairly early powerhouses which pump out units ad infinitum for the rest of the game in a few turns. Barracks, Stable, Heroic Epic, Military Academy, Military Instructor, Forge, extra National Wonder (Ironworks*/Moai Statues/Globe Theatre*, though the ones marked with an * often take too long to be practicable given that I am usually in a state of perpetual war by the time they come along), and perhaps a few buildings to help health along at the later stages of the game, and you can have a six-pop city churning out units all day long. (I rather think of them a bit like Stalingrad during the war - the people are all in rags and coughing up blood but they still deliver the goods and have one of the best morale ratings in the entire empire because they are fed a diet of propaganda /bread and circuses along with their herbal tea - good for the circulation and as a prophylactic against bacteria - and their salted sprats.)
They might not grow that much over the course of the game but they are usually well inside my empire - as the first or usually second city - and are usually sited so as to take charge of strategic resources. Because my gambit on Warlord goes Mysticism --> [Stonehenge] [second city] ---> Polytheism/Religion if poss --> Masonry --> [Great Wall] --> Priesthood --> [Oracle]/Code of Laws/Religion if poss --> Monotheism/Religion if poss --> GP for shrine --> GP for Theology/Religion if poss (though I have been beaten to the three late religions, particularly if Wang Kon lightbulbs Philosophy in the mid-BCs as he is wont to do) --> [Apostolic Palace] I make sure I have a decent production site early on and REX to more advantageous sites for population growth and commerce activity later. I enjoy playing Financial and/or Creative and/or Organized leaders, because I run a CE, and this strategy isn't affected by the new patch because although they give you more money to start off with they take it from you in unit maintenance outside your borders as soon you start the first axe rush. I know I spam cottages too much and neglect farms/food, but I also like Windmills around production cities to get a bit of a boost later on and as soon as I get Replaceable Parts the windmills kick into the production field as well as the commerce/food field.
(2) And even when you are over the other side of the board from your empire they will still show up. It is just a guideline, nothing else. If anything they are less annoying than the dialog boxes in Civ II which appear as part of the tutorial mode and infuriate me because it usually ISN'T a good site for a city.
(X) When considering a city I don't tend to look at where it is but as I enjoy financial games and commerce-economies I tend to go towards a coast at first and line my cities up to make sure I get a lot of ports. Then I just land-grab in-land. I do put my cities on empty desert tiles to wring as much as possible from that "empty quarter" but at the moment I am using the strategy layer to dotmap the whole continent and stick within what I've delineated for myself. Amazingly the AI either tends to know where I've dotmapped (one AI got its city to fit perfectly and then of course promptly lost it when I DoWed to get it back!) or tries to muck up my dotmap with spam settlements of its own (once I had to go to war against Saladin twice to get back the nice spot I'd chosen because I neglected to have a Settler ready among the axe rush

) so I'm coming to the conclusion it may not be best to dot-map in-game to avoid giving the enemies' spies the broadest of hints with a huge purple line drawn down the middle of the desert. Perhaps that's where the idea of ley-lines comes from? God trying to dot-map...