Combining CMOS and TTL with drivers and Leds with relay driver
You do not need pull up resistors when driving CMOS inputs with TTL outputs unless the TTL outputs are specifically open collector. (check the datasheet)
It should not be a problem to connect an output to multiple inputs as long as the input currents don't add up to more than the output can handle. (check the datasheet) In most cases, the input current will be under 50uA for HIGH and under 1mA for low. You do have to be careful if you are adding external pull-ups to not make the outputs sink too much current in the low state.
You do not need resistors when connecting TTL to CMOS or the converse as long as the Vcc is the same on all devices. The reason that you might want to have a resistor is when you are driving some of your CMOS at higher voltages than the 5V limit of TTL.
The NE555 has a TTL compatible output which can sink or source 20mA. There is not problem with driving multiple gate inputs directly. (check the datasheet since there are some low power versions of the 555 that may not comply with this statement)
The 74F269 will function better driving LEDs as a sink rather than a source. If you look at the IO Loading Table you'll see that the outputs are only rated to 1mA in their HIGH state, but they can sink 20mA in their low state. In the current sink configuration you would connect the cathode of the LED to the output and the anode to your current limiting resistor and the resistor to Vcc. This will allow you to run the LEDs at a much higher brightness than in the source configuration. The only issue here is that the LEDs are active low.
Make sure you don't mix up the AND (74LS11) and OR (74HC4075) in you circuit like you did in your question.
One problem that can be encountered when mixing TTL and CMOS is when running circuits at high speed, the threshold value differences can cause timing errors which may result in transient invalid outputs.