Man v Man is seen in all of the stories. Man v Self has been seen in all of the stories because the main character is always dealing with some inner conflict based on his own actions and thoughts. Is this true?
Obsession has been a reoccurring element in the birthmark, Tell Tale Heart, and The Cask of Amontillado
Obsession occurs in the birthmark through the husband trying to change his wife’s appearance. In Tell Tale Heart, it occurs when the main character is obsessed with the old mans eye rather than the old man himself which drives him to commit the murder. In the Cask of Amontillado, he is obsessed with revenge and murder similar to Tell Tale Heart.
Importantly, what "drives" the obsession of each protagonist?

Preston and Jayla 

In the Tell Tale Heart and The Black cat we noticed that there is a theme of psychological effects of guilt or sin. (Does the protagonist suffer from guilt or is he simply trying to avoid being hanged?)
Both at the end, when the narrator slowly goes insane from the violent sins committed. However, the Cask is different because there are themes more related to judgment and envy that lead to that narrators evil actions of murder.

In all 3 Poe stories we felt that there was a common theme of denial and justification of insanity and insane actions. A recurring theme of the Poe stories is that all of the narrators either committed or described actions such as grotesque *What is a/the grotesque in literature?  murder and animal cruelty, and these narrators attempted to justify these actions or deny their own insanity. This is ironic as the situations are extremely insane and full denial of insanity by these narrators is humorous. 

Tell Tale Heart/ Cask of Amontillado/ The Black Cat Common Ideals
- Man vs Man (Aim for perfection - the perfect murder in Cask only)
- Death / Murder
- Unhealthy Obsession
 
Obsession is heavily focused on within the Tell Tale Heart and the Cask of Amontillado which is different to that of The Black Cat. The Black Cat is not a darker obsession compared to the other stories and has more of an obsession to not get caught. Short but good!

Rip Van Winkel, The Birthmark, and The Black Cat all have domestic trials especially through the gender and societal roles present. The Birthmark and The Black Cat are specifically man vs. self? _{Man Vs. Man t}hroughout the whole dialogue. Black Cat and Birthmark differ within the ways the main character expresses their inner turmoil, the black cat keeps the characters Protagonist's? thoughts to himself (He has conflicting thoughts throughout the story rendering him completely unreliable, and without remorse.)
and audience while the Birthmark keeps other characters and audience informed. In all of the pieces there is an aspect of supernatural beings, spirits, greek myth, and the understanding that a higher power is present.
You should be talking about the way all three protagonist's treat their spouses. Rip - Neglect/Birthmark -Obsessive Destruction/BlackCat - Abuses suffered at random whims of narrator's impulses. 

In the “black cat” , “tell tale heart”, and the “ cash of amantillado” the stories all portray an animalistic element. In direct relationship with murder. To be classified as an animal under human definition describes a loss of reason and morality.  --
Interesting premise. This is a Man Vs. Man conflict in which Man is "prey" to a stalker. 

Julia Kedzierska
Beth Schweitzer
Adrian Rodriguez
Daniella Pecoraro

The most common theme between all the stories would be man vs. nature in a way. In the Tell Tale Heart, Man vs nature was present due to man not wanting the old man to age and he did not want the nature of his eye to exist. In Rip Van Winkle, the man preferred to get away from another man and spend his time in nature. While in nature he viewed a mythical source that took away 20 years. When he came back to society everything he dreaded about society was gone. This made the love of nature very crucial to his own sense. In the Cast of Amontillado we couldn’t find anything that was man vs nature so we kind of stretched it a bit. It was in Montresor's nature to get revenge on Fortunato because it was in his "coat of arms" to get back at him. (Interesting stretch, but nope. All the stories share the Romantic Element or Tenet that is love of and wonderment at Mother Nature, but this story does not fall into that category, does it?) His arms is ​​“Nemo me impune lacessit.”, which means no one gets away with punishing me. It was bound to happen. In The Black Cat, The man was angry with life from the start as he was bullied as a kid. He then turned to alcohol and blamed everyone but himself. There is symbolism with hurting Pluto as once he did another black cat came along with a white mark on its chest showing a beheading device. (Gallows - hanging device) He never felt guilty for his actions but rather than blamed himself. he blames everyone BUT himself. This was the story of looking at a psychopath's actions and thinking process.

Not every Romantic story is going to have all the elements in it, but nice try and very creative use of Montresor's crest. 
You are confusing the word Nature - the natural world, with nature, meaning the inner world of the protagonist.