I have an EGLR too, they are fantastic at finding the right dog for the right home imo, so allow yourself to be guided by them rather than finding a dog on their website that you like the look of iyswim.
They have loads of returning dogs atm, guess it's getting close to christmas 
I have a whippet and a longdog, both are brilliant in the house and good with the kids, fine to be left. One is a fox poo magnet so permanently stinks, the other for no known reason, throws an enormous amount of coat every six months and the whole house is covered in hair for a few weeks, just so you know there can be unexpected hair issues 
They will sleep in their own beds at night in the summer (in our room) but in the winter they insist on sleeping in our bed, they are not a breed/type for anyone who doesn't want dogs on the furniture/beds.
They are insanely loving, kind, gentle, stubborn, deaf, stupid.
They daily occasionally do wall of death runs around the house at top speed and pay no attention to humans in the way, and will use you as a launch pad.
Your 18 month old will definitely get knocked over, your seven year old quite possibly, it won't be malicious and they will be fine but will you be okay with that?
They have both caught squirrels in the past, which isn't nice to see, but is part of their nature (for some not all), they both HATE cats and would undoubtably kill one should the opportunity arise, this needs to be managed to the best of my ability. Both are totally uninterested in any birds (chicken, ducks, geese) and are completely safe around them, one is livestock (sheep, cows, deer etc) safe, one, not so much.
A greyhound is likely to be much more laid back, lurchers/longdogs depend on what has gone into their make up, a 3/4 collie, 1/4 greyhound for example, would be likely (not definitely) to be active and energetic and need considerable exercise and training. Whereas a 1/2 greyhound 1/2 whippet would be mostly asleep and occasionally running like a loon.
Just my own musings, as a distraction from writing the essay I am avoiding....