Original (see Update below)

Assume I have a data set with a low two digit number of columns (some NA/empty) and more than 100.000 rows, represented by the following example dataframe

df <- data.frame(ID = c(1,2,3,4,5), 
                  CTR1 = c("England", "England", "England", "China", "England"),
                  CTR2 = c("England", "China", "China", "England", NA),
                  CTR3 = c("England", "China", "China", "England", NA),
                  CTR4 = c("China", "USA", "USA", "China", NA),
                  CTR5 = c("USA", "England", "USA", "USA", NA),
                  CTR6 = c("England", "China", "USA", "England", NA))


df

ID   CTR1    CTR2    CTR3    CTR4   CTR5    CTR6         
1    England England England China  USA     England 
2    England China   China   USA    England China
3    England China   China   USA    USA     USA  
4    China   England England China  USA     England
5    England 

and I want to count the co-occurrences by ID/row to get a co-occurrence matrix that sums up the co-occurence by ID/row only once, meaning that no value over 1 will be allocated to a combination (i.e. assign a value of 1 for the existence of a co-occurrence independent of in-row frequencies and order, assign a value of 0 for no co-occurrence/combination by ID/row),

1 England-England-England => 1
2 England-England => 1
3 England-China => 1
4 England- => 0

Another important aspects regards the counting of observations that appear once in a row but in combination with others, e.g. USA in row 1. They should get a value of 1 for their own co-occurrence (as they are in a combination even though not with themselves) so that the combination USA-USA also gets a value of 1 assigned.

1    England England England China  USA  England 

USA-USA => 1
China-China => 1
USA-China => 1
England-England => 1
England-USA => 1
England-China => 1

Due to the fact that row count should not >1 for a combination by row/ID, this results to:

        China   England   USA 

China    1        1         1        

England  1        1         1        

USA      1        1         1

This should lead to the following result based on the example dataframe, where a value of 4 is assigned to each combination based on the fact that each combination has occured at least in four rows and each string is part of a combination of the original dataframe:

         China   England   USA 

China    4        4         4        

England  4        4         4        

USA      4        4         4

So there are five conditions for counting:

1. Single observations without additional observations by ID/row are not considered, i.e. a row with only a single country once is not counted.
2. A combination should be counted as 1.
3. Observations occuring more than once do not contribute to a higher value for the interaction, i.e. several occurrences of the same country do not matter.
4. Being in a combination (even in the case the same country does not appear twice in a row) results in counting as a self-combination, i.e. a value of 1 is assigned.
5. There is no value over 1 assigned to a combination by row/ID.

I've tried to implement this by using dplyr, data.table, base aggregate or plyr adjusting code from [1], [2], [3], [4], [5] and [6] but as I don't care about order within a row but I also don't want to sum up all combinations within a row, I haven't got the aspired result so far.

I'm a novice in R. Any help is very much appreciated.

Update

Thanks to @jazzurro for his anwer. It made me realize that the duplicates may complicate things. I hope keeping only unique values by row simplifies the task so that the example dataframe looks like this by applying code from here.

df <- data.frame(ID = c(1,2,3,4,5), 
                  CTR1 = c("England", "England", "England", "China", "England"),
                  CTR2 = c("England", "China", "China", "England", NA),
                  CTR3 = c("USA", "USA", "USA", "USA", NA),
                  CTR4 = c(NA, NA, NA, NA, NA),
                  CTR5 = c(NA, NA, NA, NA, NA),
                  CTR6 = c(NA, NA, NA, NA, NA))


ID CTR1    CTR2    CTR3 CTR4 CTR5 CTR6
1  England China   USA
2  England China   USA
3  England China   USA
4  China   England USA
5  England

Leaving only three conditions to be fulfilled:

1. Single observations without additional observations by ID/row are not considered, i.e. a row with only a single country once is not counted.

2. A combination should be counted as 1.

3. Being in a combination results in counting as a self-combination as well (USA-USA), i.e. a value of 1 is assigned.