So based on Fabian's answer here combined with Oak's experience mentioned in the comments for this question, it sounds like razing a city will cause all tiles that were acquired by that city to become neutral, as well as all tiles in the immediate ring around that city, even if those tiles were not initially acquired by the city being razed (beacuse as Fabian pointed out, building/capturing a city will automatically transfer ownership of tiles in that inner ring that belong to other cities of the same player to the "new" city). Any tiles in the inner ring that do not belong to the player owning the city should not become neutral.
Unfortunately, I am not aware of any way to determine ahead of time which city initially acquired a tile and thus "owns" it for purposes such as this.
How acquiring technologies works
First of all, it's important to understand that you do not get technologies by the number of turns you invest, but by the number of science points you have, just like it works for culture policies. It's just hard to see when looking at the technology graph because it only lists the amount of turns it will take; but each technology does, in fact, have a cost in science points. You can see that cost by opening the Technology's Civilopedia page (this can be quickly done by right-clicking the technology in the graph).
This is the reason, by the way, that you sometimes get technologies earlier or later than what is listed in the graph - because the estimation is done based on your current amount of science per turn, and that can change.
Also, be aware science points overflow - if you completed a technology, all left-over points will be immediately put to use on your next technology being researched. This applies regardless of the source of science points, so it will also work with a science boost (see below).
How the science boost works
So, a science boost is given in science points, not in number of turns. If I understand correctly, a Science Boost works this way:
- Make a list of all the technologies you can currently research (that you have all the prerequisites for).
- Find the cost for each of these technologies.
- Find the median value of the above costs.
- Multiply it by 0.5. This value gets boosted by 0.25 if you started Rationalism, 0.25 if you have the Porcelain Tower and 0.5 if you both started Rationalism and have the Porcelain Tower.
This number is the number of science points you get.
For example, say you can currently research Steam Power, Biology and Fertilizer. Steam Power and Biology cost 2350 Science, while Fertilizer costs 1600 science. The median of the set {1600, 2350, 2350} is 2350. Let's also assume you have the Porcelain Tower. The number of Science points you will get is 2350 * 0.75 = 1762.5.
By the way, on a personal note, this is a much better system than the previous one, in which you just got the cheapest technology available for free... although admittedly this is also much more tedious to calculate. As a rule of thumb, this will usually advance you by "half a circle" in your current research.
How to maximize the effects of a science boost
As for maximizing the effects, there are 3 basic ways to make it better:
- Build the Porcelain Tower, which is a great Wonder even without this boost, on account of the free great scientist.
- Start Rationalism. That's a great policy anyway for getting more science.
- Make more research agreements! Remember you can make a lot of research agreements simultaneously, as long as they are not with the same civ. Also remember that the AI civs love their research agreements, so if they won't make them with you they'll make them with the other civs - meaning two other civilizations get stronger and you get nothing. It's better for you to be one of the two.
Best Answer
The answer is: it depends. :)
The best strategy is to have a strategy, i.e. decide how you want to win the game in advance. Are you going for a domination victory? Scientific victory? Cultural victory?
Pair this with your chosen civilization and the surroundings of your city(ies) to decide what to build and research.
Are you going for a cultural victory and are playing Ramesses? You should try to build as many wonders as possible and any improvements that'll help you do so.
If you're going for a domination victory with the Chinese though, you should go for military buildings, anything that'll enhance production of great generals and happiness-boosting buildings (since your sprawling empire needs to be kept happy).
For any civilization though you need to balance these decisions with the given situation. Is your city in the mountains but has very little access to food? Then you can safely skip the production-boosting buildings but should concentrate on granaries and water mills. Or are you situated on a tiny island? Then harbors and marine buildings should probably be your preference.
You need to play to your civilization's strengths and balance out your situation's weaknesses. Playing "evenly all over the map" usually works in easier difficulty settings, but you'll need to specialize and focus on higher difficulties. If you set yourself a goal, the question of what to build or research next usually answers itself.
Try to shoot for a cultural victory for instance, and begin by choosing an appropriate civilization. This should be a good experience to show you how playing a focused game can turn out. Playing "all random" is usually more challenging, since it may take quite a few turns until you have formulated a strategy (or you may never get around to formulating one at all).
Anecdote: I just won a cultural victory with the Egyptians, and it was an enormously fun game. I had all of two cities with no military power to speak of (two Infantry units rather late in the game). My empire was tweaked to perfection though and was oozing culture (obviously) and money, which I used to keep good relations to my neighbor Wu (China), who apparently went for a domination victory. I was declared war upon twice by other civilizations (once by two civilizations teaming up against me), and both times used my good relations to have my enemies wiped out by China for me. I had to bribe my way through the last 20 or so turns, since Wu and I were the only civilizations left (~40+ cities vs. 2), and I was the only thing standing in the way of Chinese world domination.
Lots of interesting gameplay to be had in this game. :)