I'm writing from what I've observed in my daily life; it may be wrong.
Entrance & admittance are used when physical or any other specific aspect of the act (of entering) is in focus.
Entry & admission are used in more conceptual or abstract sense. These are used in situations where the fact that the act has happened is more important than the how it happened.
Entry & entrance involves crossing/going in some kind of boundary or border.
Admission & admittance involves permitting/inclusion of someone/something into some kind of structure/system or a well defined region of space with specific purpose.
Now coming to the examples:
The burglars force an entry into the rich man's house.
Here the house would be the border. Entry is used because physical aspect of the act is not important. We are not talking about how or from where the burglars entered the house, the focus is on the fact that they managed to enter into the house.
Consider this example,
Engineers blasted upper portions of the granite slab, forming another entrance into the cave.
Here the physical aspect, a literal hole in the cave wall, is in focus.
He refused me entrance to his house.
It is better to use entry in this sentence, as it is less focused on the way and more focused on act of entering (or on not entering in this example).
You can also say,
He refused my entrance to his house.
Here more emphasis on your entrance. This makes it more specific and less abstract.
The headmaster's sudden entrance frightened the pupils.
Both words can be used here with slight change in focus, as
no clear boundary
is defined in this sentence. Using entrance makes us ask questions about the place (physical aspect of the boundary). It is an examination hall or a classroom perhaps? (entrance to where?)
Using entry arises more abstract questions about the ongoing state, like what is the previous state of pupils before they were frightened. Attentively listening to class or tensely waiting for their grades. (entry to which kind of situation?)
The villain makes his entrance in Scene III.
The entrance of the pop star was greeted with shouts and screams.
In these two examples the physical aspect is more important, like the background music/shouts or the abruptness of the scene change. Here the boundary is the existing scene/composition of the stage. Entrance is always used in these kind of situations.
One is usually not allowed entrance to a room where dangerous things are to be found.
As said in one of the earlier answers, 'admitted' should replace 'allowed entrance' in this sentence. A room containing dangerous things is not just a boundary, it is a complete system/space.
Entry to a room, where dangerous things are to be found, is not allowed.
This is a better version as entry and allowed are separated; room is more represented as a boundary. But I still support using admitted.
There are signs saying 'No Entrance' everywhere.
Use entry as it represents any kind of entering. A 'No Entry' sign discourages any kind of entry or the act of entering itself; it doesn't mention about the
method of entrance or any other specific aspect of the action.
The old bridge has a sign reading "No Entrance to Heavy Trucks."
Here it is more focused. The sign cares about weight or heaviness (physical).
The refugees were not granted entry to any country.
Here, country is used a boundary, rather than a separate system. As it is mentioned in an abstract sense, entry can be used. See other variations:
The refugees were not granted entrance at St.Hans Bridge.
We are focussing on a specific place (which exists physically).
Admission of a refugee into government colleges is very difficult.
Admittance of a refugee to the president's swearing-in-ceremony is impossible.
Here we are referring to the venue where the ceremony is taking place (even though not stated explicitly).
One thing to note is the usage of admittance is decreasing recently. One reason may be because our systems are becoming more abstract with time. Physical buildings like the White House have far less significance in today's impression of a Government than a few centuries ago, when a physical objects like crown or throne were more important.
You cannot just go into the club as admission is restricted to members only.
I see no problem here. You should use entry if use club-hall or club building, instead of club (they are defined by borders).
To go into a foreign country, one usually needs to have an admission visa.
Here 'country' is generally treated as a border, so entry should be used. Admission can also be used if country is treated as a system, into which immigrants or visitors are supposed to get assimilated.
Admittance to school depends on places available.
Admission is more appropriate. School is defined as an abstract entity here. It is used in the meaning 'institution of education' rather than as the building in which teaching is done.
Admittance to Nuclear Control room needs an official id.
As N. Presley said above, admittance is better used only in the meaning of "authorization to physically enter a structure".