1. Yep, you can take that barb city, it seems good enough and you'll receive some conquest gold as well. That must be a rather old city, thus the amount of conquest gold is going to be reasonably good.
2. Agree. Delay IW at least until Calendar.
3. I'm quite sure that domination is the fastest way to win the game, but you could do something different, may be space.
4. Some cities are a bit tricky. To decide what improvements to use in a given city, you need work out what that particular city is fit for and find the best way to use it to carry out your general plan.
a) NE city. You want it to run as many specialists as possible with as greater multiplier as possible. Therefore you want Karakorum to grow fast and chop all the necessary stuff. Preferable order: granary, library, GL, NE. Don't run any specialists before NE, it will only slow everything down. And most certainly you don't want Karakorum to work cottages.
b) Agra is another potintial GP farm city, there simply is no other use for that amount of food. Of course, you could share it with another city, but lack of good tiles suggets that first option is better. With caste+pacifism its 4*3*2=24 GP points per turn.
c) Pataliputra, Ning-hsia, Beshbalik are pretty straightforward - work food, cottages and other good tiles like riverside ivory, horses, calendar resources. Don't build cottages on non-riverside tiles, they are rather bad without financial.
d) The GLH in Vijay. Yeah, you can build it. Lets see, with OR you need 200/1.25=160 raw hammers, 90 will come from chops and 14 from overflow, that leaves 56. At 9

/turn its 7 turns. Vijay will grow in 3 turns, so its actually 6 turns. Not bad, but you'll need another worker there.
Make failgold. For instance, ten chops into any marble wonder will make 30*10*2.25=675 gold, enough to sustain ~12 turns of research at 100%. Not too shabby?