I think you mostly have it right, but for further exemplification, here are the most common immediately following collocates for each these words from the COCA:
Most common collocates for mandatory
1 TESTING 151
2 MINIMUM 125
3 RETIREMENT 107
4 MINIMUMS 100
5 REPORTING 87
6 SENTENCES 78
7 EVACUATION 74
8 SENTENCING 74
9 DISCLOSURE 63
10 ARBITRATION 56
11 SPENDING 41
12 DRUG 36
13 SENTENCE 35
14 EVACUATIONS 32
15 ARREST 27
16 PRISON 27
17 COUNSELING 26
18 OVERTIME 24
19 RECYCLING 23
20 BUSING 22
21 SANCTIONS 22
22 INSURANCE 21
23 SERVICE 21
24 HIV 20
25 RELEASE 20
Most common collocates for compulsory
1 EDUCATION 86
2 MILITARY 53
3 SCHOOL 31
4 SCHOOLING 21
5 STERILIZATION 19
6 LICENSING 18
7 ATTENDANCE 18
8 NATIONAL 16
9 SERVICE 15
10 MILITIA 14
11 MEMBERSHIP 14
12 JURISDICTION 11
13 HEALTH 11
14 HETEROSEXUALITY 10
15 LABOR 10
16 ARBITRATION 9
17 LICENSE 9
18 FIGURES 9
19 PROCESS 9
20 PUBLIC 9
21 RETIREMENT 8
22 DISPUTE 7
23 ARMY 7
24 VOTING 6
25 INSURANCE 6
Most common collocates for obligatory
1 SERVICE 10
2 NATURE 8
3 STOP 8
4 REFERENCE 6
5 STRUCTURE 5
6 CELIBACY 4
7 EXCHANGE 4
8 VISIT 4
9 READING 4
10 MILITARY 4
11 QUESTION 4
12 SLAP 3
13 NOD 3
14 RETURNS 3
15 BASIC 3
16 PERIOD 3
17 UPON 3
18 PHOTO 3
19 BOOK 3
20 EARLY 3
21 PARTHENOGENESIS 2
22 PREPOSITION 2
23 PILGRIMAGES 2
24 HAGGLING 2
25 ALTRUISTIC 2
As you can see, mandatory and compulsory have much stronger collocates. The collocates for mandatory are more things that are simply required by law, policy, or rule, whereas compulsory is more for specifically things that someone must do (e.g. education, military, sterilization). Obligatory, on the other hand, doesn’t have much of a strong affinity for certain things, and can be used generally for anything that is required by social custom.
NULL is used in computing most often (always?) to signify "not a value." This is different than zero: in a bank database, a zero value means no money, while a NULL value means there has been no value assigned to the balance.
This is a specialized usage, but increasingly common as more people learn to program.
Best Answer
No doubt many people think of the two words as interchangeable, but to me backdrop (a US coinage from about a century ago, originally a literal reference to the backcloth of a theatre stage) potentially has much more restricted applicability.
In general, I expect a figurative backdrop to be something apparent to me (the observer) at the time when I'm viewing (or otherwise "focussing on") some foreground referent.
On the other hand, a figurative background might be something I'm completely unaware of until someone else tells me the backstory behind whatever we're talking about or looking at.
In terms of prevalence, even among people who treat the words as equivalent, background is far more common, but arguably backdrop sounds more artistic/intellectual/"cultured" to many.