[Phoenix-D]

"When a planet have limits in the number of facilities it can have, how I can increase the number?"

If it has a dome, yes. Build an Atmosphere Converter, or take someone else's population that breathes that type of atmosphere and bring them in, replacing the current population.

"Is it a good idea to destroy some of the facilities to be replaced by newer ones or just colonize new planet to build new facility?"

Depends on what you're replacing with what. Use the Upgrade button if you just want to get a higher level of the same thing.

"When new shipdesign or weapons available, how can I upgrad vessels? (both new ones and the ones in the field. For example, in alpha centauri, we can just push a button to upgrade all the vehicles in the field)"

Move the ship to a shipyard, then click on the icon that looks like a recycle symbol. There you can retrofit the ship into any design which is the same size- provided the two designs are similar enough.

"Now I am at the very beginning of the game and had couple of space battles. But from the ship menu I see troop transporter. thus it seems ground war is also possible in this game?"

Yup. Research Construction and you'll be able to research Troops. Then put cargo bays on your ship, set it's strategy to Invade Planets, design and build some troops, load them up and go conquer. Doesn't have to be a transport type either, warship hulls work.

Cargo bays can also be used for moving population, satelites, weapons platforms, and such.

Phoenix-D