First - are you using steam? F12 is the default for taking screenshots in steam - much easier for us to see than physical pictures.

To address your questions:

1) Roads generate happiness and gold (scales with population) if they connect the city to your capital. Each road also costs 1 gold per turn, so don't build too many!

2) First value is your income, second is the trading partner's. The owner of the trade route always makes a higher income. The hover-over is the breakdown of the calculation. You make a higher trade income based on unique luxuries, multipliers from policies, etc. External trade routes generate gold. Internal trade routes generate food or production (hammers) for the receiving city. You need a granary or a workshop for food/hammer trade routes respectively.

3) Luxuries are throughout the empire. They are useful for managing empire size, as happiness will be a limiting factor on growth. However, luxuries can only be worked locally. Only the city with the ivory nearby, taking your example, will gain the tile bonuses of the luxury. Additionally, some buildings require an improved luxury within the city's borders in order to be built.

4) Sometimes you can focus into science or gold if you prefer not to build useless buildings. To be honest, based on your progress, you may need to move up a few levels in difficulty. You've got nothing left to build simply because... well, you're doing so well.

5) Baracks (and other military XP buildings) apply bonuses only to the city they are built in. This is similar to other buildings that gran bonuses to food and hammers.

6) Science victories are attained once you build a space ship, assemble it, and launch it into space. Science is correlated to population and can be enhanced by buildings (such as the library) or social policies. Each additional city after your first city increases the cost of each technology by 5%. This includes puppets as well as annexed cities. This is why people tend to play tall (fewer but individually strong cities) as opposed to wide (many smaller, individually weaker cities).

Keep exploring, see if you can figure out some nuances to the mid-game and late-game, specifically exploration, archaeology, and national ideologies. Once you have a solid feel for what you've gone through in your first game, move on up in difficulty!