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In properly set democracy this should not be possible. President should not have that kind of power to affect other branches of power.
In general president should not have too much power at all, because it is stupid to have just one person to be able to cause so much change in general. Looking at you US.
Presidential offices and ministries are both part of the executive branch of governments, though. Ministries are primarily there to organize the executive's work, so while removing ministries will affect that, it won't affect the separation of powers (like, say, the removal of responsibilities from a court or a chamber of parliament would).
That absolutely doesn't mean that the power to create or disband ministries has to rest with the executive branch.
In fact, it can easily be argued that creating the framework in which the executive branch operates is the domain of the legislative branch - so the creation, merging, splitting or disbanding of ministries should also be a power of the legislative branch.
Or you could argue that it should be a power that should be shared between an administration and parliament, where an administration could introduce a motion to change ministries to parliament, and parliament would have to vote on it.
Lots of possibilities.
Yes, of course! To my knowledge, the creation and removal of ministries does require parliament approval in most parliamentary democracies, and I also prefer that this power is not held by a president alone (as in a purely presidential system). However, in such a presidential system, a president adding or removing ministries still reflects a change in the same branch of government, the executive branch. The process of reorganizing ministries per se is not problematic and actually required for an administration to remain functional and able to accommodate new/shifting tasks.