Converting 12V capacitors from parallel to series: do they need to be rated for 250V

You are using your transistors incorrectly. In fact, I think that the only transistor which behaves as a switch in your circuit is T2.

Note that a transistor, NMOS for example, will conduct while \$V_{GS}>V_T\$. You are biasing your NMOS transistors in such a way that the potentials at source terminals is much higher than at gate terminals.

I believe that the reason you see a constant voltage on the load is because T3 and T4 are conducting when \$VG_2=5V\$, and T1 and T5 are conducting when \$VG_2=-5V\$ (in all cases due to wrong biasing). You can easily check my guesses in simulation by measuring the currents.

Furthermore, take a look at transistors' datasheets - the voltages in your circuit exceed the max ratings for the transistors.

The component you are trying to implement is called "Charge Pump" - there are many standard implementations you can find on the internet. A short (but not complete) overview may be found here.

X and Y type capacitors (as it turns out) are types of safety capacitors, usually for higher voltage. (The X and Y letter codes can also mean a low temperature coefficient for class 2 ceramic capacitors.)

Here are the relevant points from a Safety Capacitor Basics document from Tecate Group:

• X Capacitors are also known as "across the line" capacitors. They are used between the "live" wires carrying incoming AC current. They're used in applications where failure of the capacitor will not lead to risk of electrical shock to the user.

• Y Capacitors are also known as "line to ground" capacitors. They are used in applications where failure of the capacitor could lead to the danger of electrical shock to the user, if the ground connection is lost.