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Like Cosmopolitain Cyprus, Maltese Knights, etc. which stand on Greek cultures.
But what does that have to do with culture unions being allowed to do some culture shifts that were not previously allowed?
 
I mean, I vassalized Scotland as England. So now I am thinking, if it's possible to integrate them using mission.
"Integrate" has a specific meaning in EU3 which is to annex your PU partner. What you are looking for is diplomatic annexation which just has a generic mission as sprites mentioned.
If you wait until 1603 there is a specific mission to conquer Scotland but only if they are not your vassal or union partner. That gives cores on all the lowlands provinces.
 
Why does the game keep giving me a mission to enter a royal marriage with the Golden Hoard? Is it even possible to do that? If you can, then how?
Fixed in the public beta patch.
 
I would like to ask what game settings you must use to get Ryukyu world conquest achievment? Or you can even play at easy?
Any combination of adjustments in the Game Options screen is permitted for achievements. However, I'd rather go without the achievement than cheapen such a magnificent feat by playing on Very Easy.
 
If I am of french culture (cosmo) and got french capital, and move my capital to St Petersburg and don't switch culture (yet)

Can I inherit with cores in french region then?
 
If I am of french culture (cosmo) and got french capital, and move my capital to St Petersburg and don't switch culture (yet)

Can I inherit with cores in french region then?

yeah based on primary culture group, not capital culture. So switch culture and form russia after you've inherited, if the delay is worth the cores.
 
If I am of french culture (cosmo) and got french capital, and move my capital to St Petersburg and don't switch culture (yet)

Can I inherit with cores in french region then?

I'm pretty sure the location of your capital has no effect on what cores you get for inheritance, only your current national culture (plus the HRE stuff).