[Capacitor Voltage]

In order to ensure that the capacitor will not cause problems simply read its
datasheet(of course if you can find it)

[5ohm 20watt resistor]

Simply plug in parallel 10 resistors each 2Wt 50 ohm

[Forumla for rectified and filtered voltage]

MP is right
also check the resistance of secondary winding
if it is more than 3-4 ohm you may have problems at high loads
because 7812 needs minimum 13.5-14V to work properly
dont forget to put large cap for filtering

[Attenuation]

This formula is wrong
it will not vork with your circuit
attenuation of your circuit is 40db/decade or 12db/octave
it means when you double frecuency it passes 4 time less voltage
for example its module of transfer function is
~1 at 25MHz
0.25 at 50Mhz
1/16 at 100Mhz
in your formula you must substitute 2 for n to get the correct result

[AC Voltage Regulator]

Search IRF.com
there are some application notes

[Need help!]

take a thermistor plug it into voltage divider. output of this divider will vary with temperature
then take IC LM339(4 voltage comparators. You can find datasheet in national.com) use second divider to generate reference voltages(4 resistors in series with apropriate values) and pluge these voltages to negative inputs of comparators
pluge all positive inputs to first divider
plug leds to outputs of comparators

[How this regulator works?]

Q13 Paase negative charge to filter capacitor at every pulse of supply
so regulation of charge regulates output voltage

[Convert DC motor to AC power? NEED HELP]

Check output voltage of your computer board
Then you can connect external DC power supply in series with in to increase sieed of motor
To connect it to AC simply use transformer with apropriate woltage and diode bridge

[50Hz when you touch Oscilloscope probe]

Inside Your Scope there are two capacitors vith value about 10nF connected between Mains lines and case of Scope for filtering purposes. they Create some AC voltage on Scope Common. Scope Shows voltage between common and probe. When You tuch probe You ground it by Your Body capacitance so voltage difference is created. To avoid this simply ground the scope common.Scope must have spetial coonnector for Grounding