A "Power-on reset" is a very common function, and simple to implement:
That will generate a brief low pulse as power is switched on. If you want a high pulse, put the capacitor above the resistor and diode instead. It will work equally well at other voltages, 12V or whatever.
I'm not sure what you mean by a "delay gate"??
To delay a logic signal, you can use a series resistor and cap to 0V, feeding a schmitt trigger input logic gate - a 40106, 4093, 74HC14 etc.
Use another inverter section to drive the R-C to get the signal back to the same polarity.
For resistor capacitor calculations, a very! rough but very simple approximation for "ballpark" mental calculations, as the time depends on voltage thresholds, type of circuit etc., is:
One megohm and One microfarad give One second.
So eg. if you wanted somewhere around 1mS, you could use 1M + 1nF, 1K + 1uF, 10K + 0.1uF etc.
eg. for 1/1000th the time, take three zeros off somewhere between the R and C values.
Use temporary components to see what time it really gives in the actual circuit then adjust a value if it's too far out.