Welcome to Civfanatics,
Jayhawks 

We call Vanilla the original version of the game "Civilizations 4". Be sure to play with the latest patch. You can also be interested in playing with the HOF MOD, it doesn't change gameplay at all but it makes information more easily accesible.
Then you can buy the game expansions, called "Warlords" and "Beyond the Sword" (or simply "BTS").
9 cities: Don't get all of them at the same time or maintenance will kill your economy. A quite standard expansion rate would be 4 cities at 1000BC, 6 cities at 1AD.
Happyness resources: They are all-important at the beginning of the game! You need your cities to grow to work more tiles, you don't want to be constrained by the happyness limit. It is true that, at the end of the game, you don't need them because you have raised the cultural slider. But don't neglect them in the beginning!
300-400cpt when your cultural slider is at 100% is too low. You are aiming at something like 800, 800 and 600 culture per turn in your 3 cities. You need to get more cultural multipliers: liberalism allows FS which adds +100% in all your cities and if you get there first you can choose Nationalism as your free tech, building the Hermitage which adds another +100% to one of your cities.
You are right that tech trading is the key to winning Deity games. They simply research too fast, so your only hope is to use their research capability in your favour. Try to learn which techs they are less likely to research and research those yourself, instead of the ones you really need. I think that with tech trading off it would be almost impossible to win a cultural Deity game.
As for your difficulties with research, it is hard to give a good advice without more information.
- Have you ever played a spaceship game to completion?
- You get an early Academy in your capital, don't you?
- You addopt Bureaucracy as soon as you can, don't you?
- You have your capital heavily cottaged, don't you?
Finally, and I am getting a bit personal here, why can't you win a Deity cultural game? Hmm, hard to tell, but...
- Why play on Deity? Deity is really hard. You don't need a wonderful map to be able to win on Deity, but there are some particular Deity maps that nobody is able to win. It could be a good idea to first win a Cultural Emperor game, then win a Cultural Emperor game before 1600AD, then try for Deity.
- You seem to need to improve your research capabilities. Why don't you try to play a random Emperor game. Don't complete it, just play till 1AD, your goal is to be doing 200 or 250bpt at 1AD.
- Reading about other people games can throw some light into a big defect in your gameplay you didn't now you had. Why don't you try participating in HOF gauntlets or in GOTM games?
- You might be interested in reading about other ways (without Pyramids) of winning a Deity cultural game. This thread here might be of interest
http://forums.civfanatics.com/showthread.php?t=298093
I hope this helps, good luck.