Learn English – end up doing something VS. end up with doing something VS. end up in doing something

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.

On the forum you linked to, a long catalog of uses was posted at 20-Mar-2008, 08:27. I think that posting makes clear that

1. there is no difference between the two phrases in the senses of "continuing to support/accompany/practise/adhere [figuratively] to": stick to/with the plan, stick to/with me, stick to/with your principles

ANOTHER EDIT: You may find one preposition used more often with any particular object stuck to—see bib's response—but this doesn't exclude using the other.

1. only stick to may be used to signify "cause to adhere to": stick the poster to the wall but not stick the poster with the wall.

2. only stick with may be used to signify "impose a (relative) burden upon": he stuck me with the bill or she got the fellowship, I was stuck with an assistantship, but to won't work in these.

One more "stick to" idiom occurs to me: *stick it to [someone]", meaning "inflict excessive (physical, emotional, financial &c) pain upon": They had him cornered and really stuck it to him.

EDIT: And another: Stick to [one's] guns, although it fits use 1 above, is a fixed idiom; stick with your guns would mean "continue to accompany your artillery".