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I'm struggling to understand the relativity of simultaneity and position.

If my conception and birth are separated by time but not space, a frame of reference in which my birth and conception are simultaneous should exist right?

If another observer moves in the opposite direction, will he see my birth before my conception?

Suppose we take the spacetime point of your conception as the origin, $(t=0, x=0)$, then the spacetime point for your birth would be $(t=T, x=uT)$. The time $T$ is approximately $9$ months, and we are writing the spatial position of your birth as $x=uT$ where $u$ is a velocity. The velocity $u$ can be any value from zero (i.e. born in the same spot as conception) up to $c$ (because your mother can't move faster than light).

Now we'll use the Lorentz transformations to find out how these events appear for an observer moving at a speed $v$ relative to you. The transformations are:

$$ t' = gamma left( t - frac{vx}{c^2} right ) $$

$$ x' = gamma left( x - vt right) $$

though actually we'll only be using the first equation as we're only interested in the time. Putting $(0,0)$ into the equation for $t'$ gives us $t'=0$ so the clocks of the observer and your mother both read zero at the moment of your conception. Now feeding the position of your birth $(T,uT)$ into the equation for $t'$ we get:

$$ t' = gamma left( T - frac{vuT}{c^2} right ) $$

For you to be born before you were conceived we need $t'lt 0$ and that gives us:

$$ T lt frac{vuT}{c^2} $$

or:

$$ vu gt c^2 $$

We know that the observer's velocity $v$ cannot be greater than $c$, and your mother's velocity $u$ cannot be greater than $c$, so this inequality can never be satisfied. That is, there is no frame in which you were born before you were conceived.

The rule is that two events that are timelike separated, i.e. their separation in space is less than their separation in time times $c$, can never change order. All observers will agree on which event was first. For the order to change the events have to be spacelike separated. In this case this would mean $uT gt cT$ i.e. your mother would have to have moved at a speed $u$ faster than light between your conception and birth.

Simultaneity is relative, but causality is always preserved in moving from one reference frame to another. In no reference frame can you be born prior to being conceived or be born at the same time that you're conceived.

Another way to recognize that causality is preserved is to notice that for events to have ambiguous time order (i.e. you could switch your experience of their order with a Lorentz boost), they must be space-like separated. If two events happen at the same location, their time order is unambiguous. No Lorentz transformation could switch them.

There are coordinate systems in which your birth preceded your conception. However, special relativity deals only with coordinate systems that can be related through translations, rotations, and transformations that are known as Lorentz transformation. Translations correspond to changing where the origin of the coordinate system is, rotations correspond to changing what directions the axes point, and a Lorentz transformation corresponds to changing what is considered "at rest". General relativity is more complicated, but those complications don't affect the answer to this question. Your conception and birth are what's known as timelike-separated events. For timelike-separated events, you can't switch the order through any of the standard transformations of relativity. So technically there are frames of references where your birth occurred before your conception, but those don't correspond to the frame of reference of any physical object, or possible physical object, under current understanding of relativity.