1. Yes. The initial turn time will become lower most of the time because you might chop forests, improve tiles, citizens might be born and be used for production, etc. If you have a very food intensive city with few or no hills, and you need to speed production do not be afraid to use unemployed citizen for +1 production each. It helps sometimes, especially when you don't have any more happiness to grow.
2. Yes.
The thing is, sometimes it makes sense to just buy with cash a building instead to produce it. This depends on your economical situation and on what are the priorities. There is no standard rule to what should be bought and what should be produced. Sometimes, it makes sense to buy a Market, sometimes a University and other times a Worshop (when all 3 are available, for example). It depends on what you need the most to make your empire as strong as you can.
Production order is important, it's best to build food buildings first, then production buildings, then science, and only after these are taken care of, the other stuff. There are exceptions but this is how I do it and it works very nice.
For example: granary/water mill/aqueduct/workshop/library/university/ is much better build order than, say, library/university/workshop/aqueduct/granary/water mill;
Regarding food focus: As a general rule of thumb, you can't go wrong with food. You can test this by playing 2 games, one in which you focus food like crazy and use only food caravans, versus a game where you don't prioritize food as much and use production caravans. You will notice what it means. Exceptions are: 1) when building shrine tring to rush a pantheon (you can even starve so that you build it 1-2 turns faster), 2) building a wonder - same, build it while starving 3) during World Congress Projects - starvation needed if your production isn't high enough to win confortably.
Food focus also increases city defense so it can actually mean that you don't need to buy walls, saving you precious production or gold.
I hope this helps.
