I would like to answer You because I'm weak in exactly the same categories.
1)
I never whip, in fact I even dont switch to slavery

2) I do only rough specialization of cities (Oxford, heroic, Wall street etc)
3) I dont turn any citizens to secialist,
only specialist I get, are free ones.
4) I love
superior late cities as well and I feel harassed if I need to place cities so the tiles will be blocked by each other.
So maybe this is the reason, why Im not able to beat Deity. Playing carefully selected starting locations and civs, however I can beat immortal about 33% of games and emperor about 66% of games.
Few suggestions that can improve Your games without/before learning early rushing, micromanagement etc.
1) 1st thing focus on 2 food squares near capital. If these are animal, Animal husbandry could be first tech. try to grow to level 3 ASAP. if You want to abuse AI stupidity, go take over their workers, declaring war. Doing it very early brings very little hazard of counterattack. Reduce risks even further, You can pick whom You gonna rob worker of. Peaceful civs seal peace soon.
2) if You have gold/gems near, you can wait with pottery a bit and go mining next to fund first few cities. As You wrote, You run short of money later soon. As cottages grow based on turns that You use them, try to place some cottages early. Early You basically skip turns, but Your cottages are growing already. If You are not able to finish game quickly and start dominating with power, cottage spamming would be the best option.
3) With easier difficulty, this is very easy. try to go tech path: masonry--> BW --> writing --> mathematics (grab both GE wonders: pyramids and hanging gardens, this will guarantee, You will have couple of more wonders later on). I have 4 cities when I start building pyramids in capital. I cancel forest chopping 1 turn before chopping and finish work when I reach mathematics. Good indication for pyramid building is Great Wall - if You see it built and wanna get Pyramids, You need to chop fast and will still probably get it.
4) After getting hanging gardens I love to build my cities even a little bit more. I normally go aestetics-->literature and then go for medieval techs like machinery, feudalism and civil service. By that time I wont have and wont need more than 7-8 cities, each having few cottages.
5) On Prince it is ridiculously easy: You will be far ahead Ai in tech race. On option is making City raider maceman and upgrade it rifleman later on. You will need almost no bombardment.
6) If You dont take initiative yourself and play defensive, You need catapults, to counter incoming stacks. AI units will be more experienced as he battles around. If You defend, You must seal Yourself safely. I go castles + horse archers with double reatreat, whitch will give 50% chance of retreating (read, damaging AIs siege) against almost any unit.
So even on Immortal level I dont build any musketeers etc. I have catapults, 7-8 horsemen, 1-2 pikemen, 2-3 strenght veteran macemen and i keep building city raider macemen for decisive moment, when I reach to rifling. Upgrade only few of Your army straight away (as You would like to upgrade Your core army to city raider 3)
7) Im not avoiding wars at all costs. If my neighbour gets attacked, I make sure I sneak for couple of cities from behind its back, but I dont take long crusaders before rifling, as I try to play without loading, I hate, when my army moves far, I get sneaked and couple of hours of play zeroed by Montezumas hordes.
The strenght of this strategy as I see it, is that You dont have to play "ALL IN". Even without horses resource etc. You can still have a winning game.
THINGS I KNOW I DO WRONG
First - I'm probably not aggressive enough with whipping. I generally will whip only if there is a clear advantage in doing so (saving 10 or 15 turns on a build that I want - a granary or library, or especially courthouse) and only in cities where I'm not running specialists.
Second - I'm not good at all with city specialization. I tend to try to build everything in every city, making every city look like every other - so they all have libraries, markets, etc etc - and I strongly suspect this isn't helping my cause.
Third - I'm not very good at running specialists. Sure, if I have the pop for them, I love to have them - I let my cities get to their caps and then try to run as many as I can. But I suspect there are smarter ways to manage them...
fourth (and here I know I DO just need to go read the war academy!) - I'm not very good with wars, particularly early wars. Late wars I do great with - airpower, armor, riflemen/squad infantry; I know how to handle those. Macemen and catapults - I'm not so good with. The end results are the REALLY important early wars, I tend to get swatted in - making the early game tough!
fifth - city placement. I'm kind of an orderly guy when it comes to my city placement; given a choice I love to make a perfect patchwork quilt of cities stretching across the land, where every tile is used and overlap is minimal (I'm OK with overlapping a little, but generally try to avoid more than a 2 or 3 tile overlap...) So, while I try to place cities for resources or strategy, sometimes a city gets dropped because ... well... because that's where a city needs to go! Never mind that the terrain is mountains and desert - GET OUT THERE AND BUILD MY CITY; it's part of the larger plan!