Non-isolated power supply regulated transistor LED driver circuit Reactance
Question 1, what is the purpose of R4 680? Since we already have c1 and it's 9.8k.

It is to prevent high inrush currents.  Also to prevent high currents during switching transients on the supply.

Question 2, Since there is a 1mA current limit by the C1 and R4, what exactly is the current for LED1-LED4?

$$\lt 1~mA$$

Question 3, what is the current going through D5? less than 1mA I suppose.

Correct!

Question 4, if I want to increase the brightness of these LED, which are way too dim, which component do I have to change?

Depends...

If you remove the transistor and short 3 - 1 are the LEDs bright enough?

If not, increase the capacitor C1.

Otherwise adjust the variable resistor.

Question 5, I know this is very dangerous circuit because it's not isolated, but since there is a 24 zenzer diode, so the voltage at the led leads are probably less than 24v. How dangerous is it to touch those led leads? 1-10, 10 being the most dangerous.

10 - Still extremely dangerous and can cause death.

If you want a better answer... ask a better question.