Fred
I think there is some light at the end of this tunnel...until you hit the next brick wall at least

Karen, you were spot on with that death entry for a Janet Allan McLuckie, her age shows unrecorded on the index but it is her, widow of Alexander. She died at the address that we have her on now for 1861, son John with an address in Bridgeton, Glasgow reported her death at the grand age of 95! Her parents John Allan and Margaret Simpson.
Janet Allan was the mother of your Robert. Jane Bell pre-deceased him when he died in 1894
The marriage in 1812 of an Alexander McLuckie and Janet Livingston is a bit of a red herring given the names and the location.
So, we have Alexander and Janet (Allan) born in Larbert Stirlingshire in the 1770s. We now have Janet's parents and this looks like her christening on 08 NOV 1778 in Larbert. Fred, if you do a parent search with these names there is a large number of submitted and actual extract entries for siblings.
For Alexander, you can for the time being only make best guesses at this stage. From the names we have and the children that you showed:
c. 22 May 1803 Margaret - Janet's mother
c. 14 Jul 1805 John - Janet's father
born c. 1812 Robert
One more christening shows for them:
HELEN MCLUCKIE Christening: 24 SEP 1809 Barony, Lanark, Scotland
I can't see a marriage for Alexander and Janet Allan, but given their approx. birth years, and the earliest birth you have, would look to have been c. 1800ish. You also have the first couple of children above with Janet's parents names, so there abouts I would say.
Most of the IGI entries for births of Alexander McLuckie in the 1770s are in Larbert which won't help....
Monica