First of all, welcome to Civ IV, a real Civilization and a real strategy game
Second, if you really want to learn to play and get better, spend more time here:
https://forums.civfanatics.com/forums/civ6-strategy-tips.540/
Civ IV is very very different from Civ V, and yes, much harder. However, things become easier once you learn the basics. The challenge is real, but really just learning a few things will get you going quite nicely on Noble level soon, so don't worry.
I will answer your questions, but really there is a bit to learn, so read up, check out S&T and ask question there. post a game to play with folks, and watch some LPs.
- Axeman eat warriors for breakfast and have a bonus against melee. Chariots kill Axes quite easily if you have access. But really, your problem here lies elsewhere. Why are barbs even entering your borders on easy levels? What other settings are you playing?
No , Cities do not defend themselves like in Civ V, but they do have defenses which apply to units stationed inside. Archers have the best city bonuses, but honestly in many games I never even tech or trade for Archery (on high levels), except for Horse Archers. So you need to learn the nuances of spawnbusting outside your borders. Don't play Huge maps for now or slow speeds. Expand.
- Cities have a happy cap. Happiness is not global (ugh) like in Civ V. There are civics and reosurces that help with happiness, and religion. But there is notthing wrong with cities being smaller until those kick in (4 or 5 citizens). Use slavery. Use it wisely. Don't feel bad keeping cities smaller until you have the happiness, but try to get it when you can and at least let you cap grow when it can.
Science does not equal population in Civ IV (again..ugh)
- Those circles are you citizens, and actually they work quite similar to Civ V in that you can change what tiles they work and also change what tiles between cities that overlap (and overlap is not actually a bad thing)
- One important thing to note is that what you think is "gold" on that map is not actually gold. Those coins are "commerce". Commerce if adjusted by the sliders (which do not exist at all in Civ V). The sliders are one of the things that make IV much more intricate than V. Lower your reserarch slider and turn commerce into gold.
Generally, you will run 100% research until you hit what is called deficit research which usually hits when you settle your second city. V has unit maintenance, but IV has other kinds of maintenance like city maintenance which is impacted by distance from capital. Also, civics can adjust maintenance as well.
There are ways into the gold, especially once you hit Currency ( a very important tech) but at some point early you will always be running deficit research. You can use binary research (100% tax or 0% research) for a few to turns to fund full research. This usually happens after Writing is in. Get a library or two up quick and run scientists for more static research. Use first Great Scientists (GS) for academy in your cap)
The most comparable thing to the trade post is the cottage. The difference is the cottage develops over time, so keep them working ( they only grow when worked). Ideally you want your capital to have good cottagable land (think riverside grass or flood plains) for an eventually Bureaucracy capital (bulk of research).
Note that while workers do have importance in V, they are much more important in IV. In fact they should almost always be your first build. DO NOT BUILD SCOUTS. Worker management is very important so use them wisely. Improve food specials first, mine or two, especially strat resources, and chop choppity chop.
Lastly, while there is a, generally, late game civic that allows purchasing with gold (Univeral Suffrage) they concept of purchasing stuff with gold is much much less a fact or in IV than V. Just get that out of your system. in IV, it is the Slavery civic that is so powerful. Understanding how to leverage slavery is the key to this game.
edit: One more not for now, FOOD is king in IV
edit2: Keep in mind that there is very little that translates from Civ V to Civ IV. You are learning a new game, but a game that is extremely rewarding. One of the best games of all time.
edit 3: Cause I'm incapable of stopping....
Commonly used UI mods. Most here don't play without them:
https://forums.civfanatics.com/forums/civ4-bts-unaltered-gameplay-bug-bat-and-bull.268/