Welcome to the forums
First thing i'll say is that its much easier to get help, and the help you get will be better if you post a save, or at least a few screenhots!
Some people even opt to play games and post regularly to get pointers from forum dwellers to help improve their play

Also there are loads of articles on everything from game mechanics to empire management and war in the forums
War Academy
I have a Coastal city(3rd one) with a great lighthouse, I built the Oracle, lost out on Pyramids due to resources being far away.
Wonderspam is usually a bad move in civ 4, at least when its done in an unplanned mish mash way!
In particular these 2 potentially game changing wonders;
Do you have a plan to make use of the Great Lighthouse (GLH)?
Building the GLH for just one coastal city is a big waste, where if the map allows, it could be used to solve your problem with Rapid Expansion as it will pay the maintenance for a huge number of coastal cities!
If you had built the Pyramids, did you have plan to make good use of them?
The main reason people build the 'mids is to be able to swap to the Representation civic to make their scientist specialists much stronger, but unless your intending to use a lot of specialists then this wonder will do little for you.
Wonders have a very high cost in the long run, using precious early hammers to build the Pyramids in particular can cause you to lose many great city sites to the AI.
Wonderspam has led to the downfall of many civs, human and AI alike!
My question is, is this game winnable now?
Almost certainly yes, nearly anything is winnable on noble
Old school civs I would just expand, it's a great land mass and then wipe the other Civs. Rapid expansion doesn't seem possible anymore.
Rapid expansion is still alive and still quite effective, while you do need to expand more slowly and plan ahead more than in previous civs it is possible to get 15+ cities by 1AD in extreme cases (bear in mind maps are smaller than in civ 3 now)
One of the biggest problems for people learning the game is usually not having enough workers, having citizens working unimproved tiles is a cardinal sin in civ 4
Another related issue is tech path, the techs your workers need to improve your lands are of the highest priority at the start, some of these being Agriculture, Animal Husbandry, Bronze Working and Pottery. Cottages from Pottery are a very effective improvement you can build entire empires on
Paradoxically Slavery is an effective tool to manage unhappiness and unhealthiness, there are articles on
happiness,
health and
slavery in the links
Can I keep a city happy on another continent(Distance)?
Happiness doesn't change with distance, maintenance increases however. How far are the other 'continents'?
Even 1 tile islands count as continents, with big trade bonuses to work with the GLH
Or should I just go to trade for now?
On noble the AI rarely gets anything worth trading for unless something incredible has gone wrong
What strategy would you use at this point.
What would an experience vet do?
Couldn't say much else without a save or pics