hey joe - i'll add my 2cents to these questions
I'm not quite in touch with your current position, but if you've got Construction then you definitely want to be using cats to take cities.
This a good question and problem one of the main concepts that newer/struggling players have a hard to grasping, as well as being a major roadblock to their progression.
First of all, as to razing cities, that is obviously situational. Certainly keep caps which are almost always good cities. If at all possible keep cities that are just plain good - good food and good resources. Raze crap cities that just won't contribute. AIs do have a tendency to settle crap cities. Make sure you get trade routes setup asap to all conquered cities.
However, back to the concept of maintenance, keep in mind that it is "relative". What I mean by that is you have to take into account all factors present in you current situation. The fact is maintenance cost are going to go up with more cities, more citizens, and certain civics. That's a constant.
However, newer players seem to fret tremendously when they start to see the slider going down. As I've mentioned to you more than once, you don't have to keep your slider at 100% or anywhere in between. If I'm conquering early, after attaining the techs I need to conquer, I will shut down research indefinitely. Relying mainly on some scientists specialists to keep some research in play. As you conquer cities you get gold from that. Then you can raise your slider back to 100% to get the next important tech like CoL.
Do you have Currency? If you have Currency you generally don't have money issues. Just build wealth in cities that don't have pressing builds. Simple as that.
The bottom line though is that if you are in positive gold at 0% research, then you aren't in trouble.
Well, the only time emancipation should come into play, especially on lower level is if you are going for space wins, maybe diplo. Again, keep in mind that you have been playing very sub-optimally in the past. Even your military victories are achieved far later than normal. That is why emancipation is in play in your games. As for later games like space, well sometimes you might have to switch to emancipation then. Not so big a deal except that in late space games I usually like to be in Caste at that point running Rep and lots of specialists. There are ways into happiness to counteract emancipation, including raising your culture slider a bit.
Diplomacy is often one of the harder concepts to master. The key is understanding the various ways to boost diplomacy with the AI. I'm sure there are things you don't realize. Also, making sure you understand who your rival is and doing things to make sure other AIs like you more than him/her.
Positive Diplo modifiers:
1) Open borders
2) Fair Trade - up to +4 (i'm sure you've seen that)
3) Shared Religion (amount varies among leaders based on how they value religion)
4) Favorite Civic (both you and the AI must be in that civic)
5) Resource trades (over time you can get up to +2 from that and it is very easy obviously...you should always be trading resources. I'll even give away single copies if I don't need them presently. Great thing to do early with EXP or Char civs since they have extra health and happiness already)
6) Shared Technology (gift enough techs and the AIs start to appreciate. You can get at least +2 from that)
7) Giving into request/demands
8) Mutual Wars (bribe your friends into war with the enemy....this can get a huge bonus with some AIs...think warmongers)
9) Liberating cities (plant crap cities within 9 tiles of an AIs capital and you get a Liberate city bonus. You can keep doing it to. Or capture an AIs old city that another AI captured and liberate it back to them. )
10) Vote for them - this is temporary though.
There may be some I missed. It's important to start to understand the differences between the Leaders. As you've probably learned, they are not at all alike. Some value religion more. Some value war. There's a lot of different parameters about the leaders.
One thing I fear (that has happened before) is getting into a war and turning down the research slider only to have another AI out-teching me. I know this is a result of getting bogged down in protracted wars- I have improved my technique since then.

I remember trying to win a diplo victory once and I gave away everything I had, but still couldn't get the last AI to vote for me.