When decline is used as a transitive verb, it means "to refuse" or "to say no to": We are declining your loan application. - I regret that I must decline your invitation. Declining a customer would be a bad business move; declining your customer base is simply ungrammatical. Probably bad business too.
When a sentient actor (a person, a corporation, an intelligent animal) is the subject of decline in an apparently intransitive sense, there is generally an implied object; I would call this a "virtually transitive" use: I offered him a job, but he declined (the job). - We offered the chimp a banana, but she declined (the banana).
When a non-sentient noun is used as the subject of decline, it means that that thing/resource/quality is becoming less, or less powerful: The puma population has been declining for the past few years. - Hari Seldon says that the Empire is declining.
When a thing is declining, or a person's health or power is declining, we can say that that thing or person is in decline. As soon as his team started losing, he went into a decline. - This country's been in decline ever since they raised the drinking age.
When decrease is used as a transitive verb, it means "to reduce the amount of": I'll have to decrease my donut intake, or else my chair will break.
Sentient actors don't decrease intransitively; you can't say He decreased.
When a non-sentient noun is used as the subject of decrease, it means that that thing/resource/quality is becoming less: The puma population has been decreasing for the past few years. but NOT Hari Seldon says that the Empire is decreasing.
A crucial difference between decline and decrease in this last case is that decline can be used to indicate a loss of power, influence, significance, etc., whereas decrease can only be used for a reduction in quantity. Thus you can say both The population is decreasing and The population is declining, but while you can say The Empire is declining, you cannot say The Empire is decreasing, since there's still only one Empire.
Vanish has a sense of suddenness or mystery or finality about it that disappear lacks.
In your examples, I would use "disappeared into the house." instead of "vanished into the house." The other two "vanish" examples are fine.
I think if you were to change your next example to, "The plane vanished behind a cloud." it would imply that it was never seen again. Disappeared doesn't have that sense.
In your last example, you could say "Her nervousness vanished once she was on stage."
Best Answer
OED
Postulate: A fundamental principle, presupposition, or condition, esp. one assumed as the basis of a discipline or theory; (also) a proposition that is (or is claimed should be) taken as granted; esp. one (to be) used as a basis for reasoning or discussion, a premise.
A postulate is accepted as true, and it doesn't need to be proven when used as the basis for another argument - it is a fundamental principle and we don't want to be re-inventing the wheel by proving it all over again
OED:
Posit: To put forward or assume as fact or as a basis for argument, to presuppose; to postulate; to affirm the existence of.
A posit, in contrast, is assumed on the basis that it will (hopefully) prove to be true. A possible explanation of how something happened is a posit. If you observe (for example) that economic inflation is occurring, you could posit that increasing wages is driving it, and then set out to collect facts to prove or disprove that posit.