So I use one power supply that powers both the chip and lights. The power going into the chip is limited by a resistor leading into the zener diode.
No.
The chip does not eat all the power that is available. It uses a very small current. The resistor feeding the zener diode provides about 5mA to a 5mA zener diode and a few mA to the third transistor.
I had planned on using 10 sets of 5 LED's (50 total) in series. Will the chip you mentioned handle the output to the transistors since I have drastically increased the current needed? Are the calculations you made for the resistor into the transistor the same for the other three into the other two transistors?
The transistors do not care about LEDs, instead they care about current. The datasheet for almost EVERY transistor shows that it saturates well (turns on) when its base current is 1/10th the collector current.
Each series string of LEDs uses 20mA and since there will be 10 strings then the collector current in the output transistor is 200mA.
Then the base current of the output transistor is 20mA which is also the collector current of the middle transistor.
Then the first transistor (previously called the third transistor) will have a collector current of 2mA and have a base current from the Cmos 555 of only 0.2mA which it can easily provide.
It is easy to calculate the resistor values.
Did you properly calculate the power supply voltage?
If an LED has a typical forward voltage of 3.2V then its maximum might be 3.6V and all your LEDs might be 3.6V. Then 5 LEDs in series need 3.6V x 5= 18V plus about 4V for their current-limiting resistor so the total supply must be 18V + 4V= 22V.
The forward voltage of the LEDs might be less than their typical voltage and might be 2.8V so calculate the value of the current-limiting resistor with a 22V supply, five 2.8V LEDs and about 25mA.