If a planet is tidally locked and is being terraformed, with the technology of 3303, will it begin to rotate to have a night day cycle, or remain an eyeball planet?
With the technology available to humanity in this game it would remain tidally locked. If it is tidally locked to its parent star, then it would have a more habitable band around the equator. The amount of energy you'd have to put into a hablitable-sized world to make it spin is beyond human means in 3300. Not only that, unless you also remove the conditions that caused it to be tidally locked in the first place, you'd have to keep applying force to keep it spinning.
For reference, the angular moment of the Earth's spin is of the order 10^{34} Nms. Lets just say thats a big number.
How long does terraforming take? And are we able to find out how long a planet has left?
Decades or centuries.
Even then. it doesn't simply "finish" - terraforming effort would have to continue even when the world became notionally Earth-like.
The planet can be assumed to be in a natural equilibrium state before terraforming started. By terraforming, you are moving it out of this equilibrium and towards another desired state (Earth-like and habitable). Unless you are very fortunate, the target state won't be an equilibrium for that planet's physical properties (orbit, mass, composition, star), and if terraforming simply stopped the state would start to change - not necessarily back to its original conditions.
Of course, given sufficent technology and energy you could _change_ the physical conditions for the world - moving it to a more habitable orbit for example. But that's not really in the realm of human technology.
Can we bring in specific commodities and speed up to the process of terraforming?
I would expect worlds in the process of being terraformed to have a greater demand for some commodities. However given the scale of such an engineering task it would be difficult to envisage a "community goal" around this.
However, there is no reason why a progress counter couldn't exist for each world, as part of the background simulation. But don't complain about grind if you set it as your personal goal...
Will population increase in a system with a newly terraformed planet?
That would be nice.
This would be by migration principly - so a corresponding decrease elsewhere. Growth through birth would be much slower - of the order of 2% per year if we look at current global population growth.
If local population _matters_, then this would bring up interesting demands in the background simulation. Why do people want to move from one place to another? What can minor factions do to encourage them to stay or come (and then stay)?