I'm about an hour away from home now.
Your close.
Pin 5/8 of the LCC110 has to go to ground.
Usually an IC with multiple gates use U1a, U1b, U1c etc. U1 is the IC # and a-f are the 6 parts.
Gates depending on the family have different characteristics especially asymetric source
And sink current. LED's should not see a reverse voltage although 7 V. is usually OK.
Sink current is usually higher. Circut would be virtually the same except for pin numbers.
There are 7 drivers in the ULN2003 and 6 in the 74HCT05.
Can you upload a pic of the LCC110 that you embedded in your drawing?
If you use a 74HCT05 it HAS to be surface mount. If we use a ULN2003 it can be a DIP
Package. Your choice.
So basically the inverters need to only SINK current rather than source it to drive LEDs and relays.
The ULN20003 is a 7 relay driver with an interesting property. If an input is unconnected, the "relay" is off. I've used this part a lot.
Layout can change a design. E.g It might be easier to route and extra gate and resistor because resistors can jump tracks.
The scanned scribble I did had LED indicator drivers (3 gates and a resistor). It had. 2 gates and 2 resistors for the LCC110 which can be simplified to 1 gate and 1 resistor.
Sorry for the confusion. I tried to download LTSpiceIV today and got an error: This is not a WIN32 application.
I may be back at this in an hour, although it's bedtime.