My advice is from a SP experience but in general things are similar as this is about maximizing yields.
- I prioritize farms on tiles adjacent to rivers/lakes
This is a good idea because you need the growth. Of course get the luxuries, strategic and bonus tiles improved first.
- I prioritize mines on hills with any terrain
You should build farms on river hills, 2f2h is a lot better then 3h. You can work the river hill even while you are focusing on growing (almost all the time) so in the long run it's a lot more hammers. Also by the time you get Chemistry all those hills should be improved, but in that case it may or may not worth it to switch to mines.
- I prioritize lumber mill on forest, i never chop it down
This is generally very bad, those early hammers are a lot more valuable than working a 1f2h tile (by the time it becomes 1f3h it doesn't really matter). It's best to chop them as a boost to critical projects like settlers (so you stay a lot less in that no-growth mode), wonders (to get them faster, or in some cases simply to get them at all), libraries in expos and of course national college. So unless you are Iroquois chop, chop, chop (but even then you still want to chop a little). The one exception is forest on tundra.
I usually chop all forests in first 3 rings, but sometimes I go for the 4th also (especially if I have Pyramids). If there aren't many source of hammers I may keep forests in outer rings.
You should try early chopping once to see how much it improves the early game (and from there the entire game).
- I prioritize trading posts on low yield tiles (tundra, desert, plains with no access to fresh water)
Low yield tiles should be worked last, so they should be improved last. Generally it is better that a low yield tile will support itself so it should produce at least 2 food. Plains should get farms, and even tundra should get farms but I would avoid working empty tundra at all. Flat desert without Petra should get nothing (except kabash if you play morocco). You are better off deleting the worker and saving the maintenance cost.
Later on, you when you have grown and the money to hammer ratio for rushing is better you can switch some of these tiles to trading posts, but by that time you probably have Fertilizer so dry and wet tiles are the same. It all depends on how much you want to grow.
- I used to build great person tile improvements on tiles with low yield but in my last game, I bulb'd them and it really worked for me
Generally it is better to bulb GP, except for maybe 1 academy (more with Babylon or Korea) and holy sites. If however you do want to plant, pick a high yield tile that you would want to work all the time: cow, deer, horses, bison. If not, at least plant on grassland so that tile can support the citizen working it. Avoid planting on river grassland as you lose too many yields. Plains is another option if nothing better is available.
Be careful when planting GP in the industrial era if you don't have archaeology. If a player has archaeology the ruins will be revealed. When they first appear they don't spawn on existing GP improvements, but if you don't have archaeology yourself you can plant the GP on a ruin and not know it. This means that anyone with Open Borders can come and completely remove your GP tile if you don't protect it.
- I really don't know what to do with jungle so I build trading posts on them
It takes 14 turns on standard speed for a worker to put a TP on jungle, that is a lot. Chop jungle on hill for mines, and river jungle for farms. Later in the game when there aren't any things to do, you can add the TP on the jungle so you can get a decent tile to work. It sucks that plains is always under jungle, but it is still the best option to chop hills and wet tiles.
- I usually play tall and build almost all buildings in all cities. I read that you should only buy necessary buildings but I have never had problem with gold (usually 250 gold income with reasonable army).
Having all buildings in all cities is not very optimal, you need some buildings in all cities (science buildings, granaries, aqueducts, workshops, markets, monuments, etc) but in general you need to carefully evaluate each building and see if it really is worth building.
For the situational buildings like stables, seaports, forge you generally need at least 2, preferably 3 tiles that are buffed in order to make them worth it. Forge is an exception if you plan to build lots of units as the buff to unit production makes them quite good. From what I know forge also buffs production to civilian units like archaeologists and caravans (this is also valid for seaport with cargo ships and work boats, but that is a lot less useful).
Stone Works and Circus should always be built for the happiness. Water Mills are also top priority for river cities. I don't really know about gardens, I usually build them in capitals, in other cities I do them if nothing better is available. If they give you an extra GS then they may well be worth it but I suspect they only give it slightly earlier, and since it's best to wait and bulb them later it really doesn't matter. I think they are a lot better if you stack them with other buffs like Porcelain Tower, Humanism, Science Funding from world council and the one from order/freedom.
Also keep in mind that not all National Wonders are worth it, it's all about the opportunity cost.
- I use the trick with production focus in citizen management and manually assign working tiles to high food tiles.
This is good, but remember to do it as forgetting to switch citizens after growing can have disastrous consequences.
- My starting tactic is usually great library rush, is it worth it?
Almost never. Unless you already have an empire set up with cities and workers, you shouldn't be going for wonders. There is one exception to this, if you go liberty you can afford delaying settler production until you get collective rule (but only if you rush collective rule) so you may be able to squeeze a wonder like Pyramids or ToA in there. Even in that case GL is probably not really worth it.
If you don't do that you should not build any wonders until you have at least 3 cities settled. By rushing GL you get one free tech but you are behind in number of cities, so you produce less yields. This will create a gap between you and the others who didn't rush wonders, and in most cases that gap only expands even though you might be tempted to think that that free tech will help you catch up.
This may be the main reason you are facing difficulties in defeating your friends. You can try not to rush it and see the results.
- I recently found out that I usually finish tradition/liberty quite fast before rationalism unlocks, what policy should i have or what to do to don't get in this situation?
This is entirely situational and depends on several factors. How many SPs do you estimate to get before being able to open rationalism? What is your main strategy? How's the current situation? etc.
For one SP the best "filler" is probably Patronage opener, but in some cases Piety opener can be quite good (especially if you have a temple UB).
For 2 SPs there are many combinations:
-you can get Honor with warrior code to have a big number of generals. In MP this may help you a lot if you plant citadels in good spots.
-piety with organized religion has a nice synergy.
-patronage with consulates for influence boost on all CS is quite good, this basically gives you 25 more turns or allied or friends status for all quests you perform after it kicks in. You need like 16 turns for it to kick in.
-commerce with wagon trains is also a solid choice for warmongering as you will build lost of roads (and maybe even send an occasional caravan as they are harder to pillage)
-exploration with maritime infrastructure is quite solid if most of your cities are coastal. At that point in the game it's somewhat like a workshop, so quite strong. Another advantage is that later you can also get naval tradition if you really need the happiness.
-you also have the option of two openers, like piety+patronage, but generally that is weaker then other combinations
At 3 filler SPs before rationalism there are even a lot more possibilities, but I will mention the best 3 in my opinion:
- Commerce up to Mercantilism (cheaper everything especially with big ben; you also get a small number of beakers which are not to be ignored)
- Piety up to Theocracy (more faith and a buff to gold, solid all rounder)
- Patronage up to Scholasticism (in MP this may be weaker as it's harder to keep alliances but you do get some science for it, and you can get CS easier with money).
If you have 4 filler policies it means you are probably teching too slow, but you can try for reformation if you have a religion, it might be the best choice.