I’m a relatively newbie too, but here’s my take:
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The text of the CCP reference sheet is “. . . If the roll is less than or equal to the total number of land zones the CCP Possesses, all remaining land zones and military units of that warlord align with CCP immediately.” That indicates that all the zones (plural) belonging to that warlord join the CCP.
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Do you mean ‘peacetime income’? If so then the UK income level is immediately adjusted on the tracking chart, but they don’t actually receive any cash in hand yet.
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Unless I’m missing a nuance the answer is the same as above. The ‘(once)’ on the reference sheet is merely to differentiate a Germany declaration of war against France (an event that can only happen once), vs something like Japan building a battleship (an event than happen multiple time and which will each earn an income increase for the US)
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My understanding is that they are separate combats (but could kinda go hand in hand). During your combat movement phase you would move your carpet bombers in to to the targeted zone. Additionally you could move in invading ground and air units. Two separate combats are in play. . . but the attacker gets to decide which one takes place first. (rules section 9.1 “The Attacking player decides in which order the declared combats are resolved.”) Theoretically you could do the carpet bombing AFTER the invasion, but that would be pretty silly.