Welcome to CFC, Mister V!
Oh boy, what a lot to tackle! Let's get started...
First off, the "60% rule" states that you expand until you must place the research slider below 60% to break even, which usually happens in the early game after you've founded around 4 to 6 cities. But if you look at the guide again, it clearly states that once the slider inches up past 60% again, it's time to expand and either found more cities peacefully if land is still available, or through war if it's not. It's extremely difficult to win the game with just 4 cities.
Don't be afraid to have cities overlap. Found cities not based upon their distance from one another but based upon the tiles, especially the resources, that they can claim. Founding early cities too far apart increases your maintenance costs at the worst time in the game to do that, because you have so few methods of offsetting it.
DO NOT AUTOMATE WORKERS. They will not improve tiles in an optimal way--far from it. In particular, one of the keys to the game is city specialization, which requires you to improve tiles around the city to support the chosen speciality (farms for a GP farm; cottages for a commerce city; farms, mines, watermills, and workshops for a production city). You need to micromanage them, sorry. Live with it.
You don't always have to chop forests to finish a wonder, but it certainly helps, especially for the early wonders where competition is more fierce. Try to combine chopping with other game elements that multiply the hammers from losing that forest (e.g. running Organized Religion with the state religion present in the wonder city, possessing the wonder's accelerating resource such as marble or stone, having a forge in the wonder city, having knowledge of mathematics, etc.).
Refer to the guide's section on happiness (and health) for additional ways to increase your cities' caps in these areas. Also have a look at Cabert's excellent articles on these topics in the War Academy (Ways into Happiness, Ways into Health).
GP are enormously helpful, especially for "lightbulbing" technologies, but they have other uses as well, yes. I usually want one of my first cities, either founded or captured in an early rush, to become a GP farm ASAP.
The best way to fight culture is to obliterate it, i.e. capture or raze the city that's giving you grief.
The religious situation will depend on a number of factors, but especially on the map. If the map gives you more landmasses separated by ocean, you can usually expect to see more of a mixed bag in this area as each continent founds a religion and has it spread. With fewer landmasses, usually one of the earliest religions spreads like wildfire and dominates.
DPs become available with Military Tradition, IIRC.