Electronic – How to output an amplitude of 10 volts PWM signal

Have a look at the specs for your servo. It will expect to see a pulse every 20mS or so. The duration of the pulse will determine the angle. N.B. This 20mS timing may vary depending on the servo type used. The length of the pulse within this time will determine the position (min about 1ms, max about 2mS)

The three connection wires are Power (+5V), Control Signal and Ground. Check the maker for specific colour code. Test the servo is working by setting up a 1.25mS pulse (0 - 5 - 0) every 20 mS. The arm should swing to the 0 degree position. Then change the pulse length to 1.75 mS and the arm should swing to 180 degrees. Putting a constant high signal on the input is not recommended.

As regards testing the output you could try a high impedance speaker (>64R) connected in series with a small capacitor (say 0.1 uF). connect one side to ground and the other to a probe (piece of wire). You should hear a low frequency hum if the output line is switching on and off.

To answer all your questions:

1. See the explanation by @JonRB, in short saying: You are making a wave going from 0V to 3V with the arduino, because the arduino does not possess a negative rail to work with. So you need to make the signal relative to a ground, in stead of its current DC offset of about 1.5V

How do you do that?

You can do two things:

1. A: You can generate a virtual ground, at about 1.5V and relate all your measurements to that and have your diode conduct into a resistor connected to that virtual ground to make it act as a rectifier. Note, however that you'll still lose about 0.7V off the signal, because that's the voltage the diode drops, or "needs to work".
2. B: You can decouple the DC away from the signal with a capacitor:

^{simulate this circuit – Schematic created using CircuitLab}

The resistor is there to pull the capacitor on the other side to a DC level of ground. The capacitor doesn't allow DC voltages to impose anything on the other side, but to a moving signal it becomes a frequency-dependant impedance. Note, that this will dampen lower frequencies, because the impedance of the capacitor is higher at lower frequencies. The larger your capacitor is the better lesser it will inhibit lower frequencies, so if you can get 10uF or 47uF. If you want 50Hz to be the lower limit (for audible for example) the drawn values will easily suffice.

2. You need an amplifier for that. For such large increases you should look into audio amplifiers or power amplifier design manuals. But I'd advice the latter only if you have a very firm grasp of signals, frequencies, transistors, op-amps, capacitors and inductors. Alternatively you could increase the voltage you are switching with your PWM stage. For this you need an external power source at the voltage you want your peak-peak swing to be and a decent transistor half-bridge (BJT, MOST or IGBT) rated at the working voltage and currents.

3. That depends on the circuits you want to power and what for. I'm getting the impression that you want to build some sort of Power Frequency Driver/Chopper. I am not going to re-itterate a full how-to on building those, but you should really consider googling those if you want to not do something very dangerous (in terms of damage or health). Also note this is a difficult and long road compared to hooking up an Arduino to a set of filters.

4. See point 2: Amplification before or after AC filtering.