Count all binary relations
APL (Dyalog Extended), 99 93 bytes

r←0∊0 0⍉⊢
R←r~
s←⊢≢⍉
S←1∊⊢∧⍉
t←0∊⊢≥∨.∧⍨
T←1∊⊢∧∨.∧⍨
{×⍵:+/∧⌿~(⍺,'0'){(⍎⍺)⊣⍵}\⍵-⍛↑∘⊤¨,⍳⍵⍴2*⍵⋄1}

Try it online!
-5 bytes thanks to @ovs for spotting R←r~, and -1 by taking the boolean negation of the intermediate results.
Takes the condition as a string, using the abbreviation as follows:
r: reflexive
R: irreflexive
s: symmetric
S: asymmetric
t: transitive
T: antitransitive

For example, reflexive + symmetric + transitive is given as 'rst' and &quot;no condition&quot; is given as ''.
How it works (before golfing)
⍝ Each of rRsStT takes an adjacency matrix and determines if the condition is met

⍝ reflexive: the diagonal is all ones
r←∧/0 0⍉⊢
⍝ irreflexive: the diagonal is all zeros
R←∧/0 0⍉~
⍝ symmetric: transpose is equal to self
s←⊢≡⍉
⍝ antisymmetric: each position in self and transpose cannot be both ones
S←~1∊⊢∧⍉
⍝ transitive: self includes the transitive square of self
t←~0∊⊢≥∨.∧⍨
⍝ antitransitive: each position in self and transitive square cannot be both ones
T←~1∊⊢∧∨.∧⍨

⍝ main function
{×⍵:+/∧⌿(⍺,'≡'){(⍎⍺)⍵}\⍵-⍛↑∘⊤¨,⍳⍵⍴2*⍵⋄1}  ⍝ ⍵: n, ⍺: conditions
{×⍵: ... ⋄1}  ⍝ Take 0 as a special case, always returning 1
,⍳⍵⍴2*⍵       ⍝ Generate all length-n vectors of 0..2^n-1
⍵-⍛↑∘⊤¨       ⍝ Convert them to n×n binary matrices
(⍺,'≡')       ⍝ Add a default function that always returns 1 to the list of conditions
{...}\        ⍝ Take outer product of conditions and matrices...
 (⍎⍺)⍵        ⍝ Eval the condition function and apply it to the matrix
+/∧⌿          ⍝ Count the matrices which satisfy all the conditions