You might want to do a walk through here with one of your games - it helps to get good advice. It's hard to give you advice without seeing a game, especially since it appears that you are using some strategies from this forum and articles. I agree that it's probably how you are putting things together.
Oftentimes the advanced strategies are not even necessary on settler, but still a good thing to practice on a lower level to get used to it before moving up. I suggest posting your game here as a walkthrough with a particular leader, strategy and victory condition in mind. The experts will be happy to give advice and point out what you may be doing wrong.
Early rushes are not critical to winning, especially at lower levels, but I still try to do them when I can. Generally, you do this if a rival is close by - say within 10 tiles. "Early" is very early. Some people even do warrior rushes. I tend to wait for either when the UU comes online(depending on leader) or axes. The key is to catch the AI offguard and beat them resource/tech wise. Steal workers and destroy strategic resources like horses, copper to prevent them from building advance units. I don't have an exact date but I'd say you would start on this well before or around 2000 bc - maybe sooner depending on speed. Otherwise, it's not really an "early" rush.
You might want to try playing at epic speed for a while.
I can't mention enough how important workers are in this game. You've probably come across this point before, but make sure worker is one of your first builds, if not your first. The general rule of thumb is 1.5 per 2 cities, but more the merrier. Build them and steal them when you get the chance. When I settle a new city I usually build a worker first there as well, unless it's coastal and there is seafood. Work you improved tiles first and get them online.
Settle commerce focused cities with cottages, golds, gems, etc, and lots of cottages. Make sure you settle one city early with high production (hills for mines) to crank out units constantly (mixed in with military focused buildings). One of the major things I've learned is a good army is a plus always, even in peace. Have the city send units to protect other cities, but also established a well-balance "Stack of Doom" that is ready to either defend or go on the warpath at a moments notice.
Diplomacy is one of the hardest things for new players and I'm still getting used to it myself. Generally allies fall along religious lines. Depending on the leader and strat, it may be wise not to jump at a religion until your rivals flesh themselves out. Who are the more powerful/advanced civs? What religion is more common, i.e., are more civs under one religion? Once this is established, choose allies wisely and shun ones that you expect to destroy.
Sounds like you should focus less on GP farms at this point and more on establishing commerce cities for research and production cities for units. While no GP farms, you can at least choose a high food city with a library, possibly your cap, and work a couple of Great Scientists. Another important point is not to overbuild in every cities. Granaries, Forges, and Courthouses are a definite in every city. However, you don't need barracks/stables in commerce/research cites or libraries/markets in production cites. (at some point later you may be able to fill these buildings in as needed). Just make sure you get the key buildings first in the right cities.
Slavery and chopping are great in the early game. Use them! The key is to get a quick tech and military lead in the early game which should be quite easy at settler level.
I'll stop here since, again, a walkthrough is helpful. There are so many tips to give and I'm not even near the expert as some folks here.
Edit: One note on cottage cities. Work those cottages! Don't worry so much about production or building. Key buildings (granaries, courthouses, forges) and rearch/gold multipliers like markets/banks can be chopped or whipped. (Granaries allow the city to grow back quickly after whipping) Just make sure you have your citizens working cottages so that the grow, which takes a lot of time. These cities will become strong as the game progresses, providing lots of beakers or gold as needed. You defnitely want lots of beakers early to get the tech lead and achieve tech goals like Civil Service, Currency and Liberalism. Another powerful things in building wealth in high hammer cities (only hammers go to building wealth, research or culture so you'll need to use a higher hammer cities for a while if they have nothing better to build or u are not a war). Building wealth allows you to keep the research slider as high as possible during the early and mid-game. This is why Currency is often a coveted tech.
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