How do the Properties of Relations work?

I'd like to change the notation of your definitions, since $R$, $S$ and $T$ would usually be used to stand for the relations themselves (and $x, y$ and $z$ would be more commonly chosen for the objects that might bear the relation to each other).

Reflexive - For all $x: xRx$

Example reflexive relation: $xRy$ stands for '$x$ is a factor of $y$' (in the set of natural numbers)

Symmetric - For all $x,y$: if $xRy$ then $yRx$

Example symmetric relation: $xRy$ stands for '$x$ and $y$ are $2$ metres apart' (in the set of all people in a particular room)

Antisymmetric - For all $x,y$: if $xRy$ and $yRx$ then $x = y$

Example antisymmetric relation: $xRy$ stands for '$x$ is a factor of $y$' (in the set of natural numbers)

Asymmetric - For all $x,y$: if $xRy$ then not $yRx$

Example asymmetric relation: $xRy$ stands for '$x$ is taller than $y$' (in the set of all people)

Transitive - For all $x,y,z$: if $xRy$ and $yRz$ then $xRz$

Example transitive relation: $xRy$ stands for '$x$ is taller than $y$' (in the set of all people)

I think you're thinking about this in the wrong way. Properties don't "work", properties are things that are true for the given relation.

For instance, you say that the a relation has the reflexive property if it satisfies the condition that all elements are related to themselves. Similarly, it has the symmetric property if for all a and b, if a is related to b then b is related to a.

You could simply say that "∀a : a R a" each time, but because this is something that happens often, this particular property has been given a name, i.e. reflexivity.

Certain types of specific relations are also given names. For instance, a partial order is reflexive, antisymmetric and transitive, and equivalence relations are reflexive, symmetric and transitive. Again, these are just names that aren't strictly needed, but they make it more convenient talking about these kinds of things.

There are several important kinds of relations, each of which satisfy a different collection of properties:

Equivalence relations: These are reflexive, symmetric, and transitive. Essentially they're relations that "behave like equality." The most important elementary one is "equivalence modulo m," where say 1 = 6 = 11 modulo 5.

Partial orderings: These are reflexive, transitive, and (anti-symmetric or maybe asymmetric, I'm having trouble parsing your logical statements). Essentially they're relations that "behave like less than or equal to." An important elementary example is "divides" where we say a|b if the ratio b/a is an integer. Note that 2|6 but 6 does not divide 2. However if 2|4 and 4|8 certainly 2|8.