What height are lumens measured at when calculating lighting layout?
Where are we getting all of these lumen questions all of a sudden? 

So let's start with your nomenclature, you start out with lumens but what you are looking for  is footcandles, which you actually reference. Lumens are the amount of light created by a lamp. Footcandles are the light measurement on a flat plane.

By reading your third paragraph you are using an R40. So I am assuming you are using a recessed can. A recessed can is designed to direct light in a certain direction. So depending on the lamp and can it could be a wide flood, a flood, a narrow flood, a spotlight, a narrow spot or a very narrow spot. This doesn't include asymmetrical designs. Regardless you need to layout recessed lighting so you don't get dark spots or shadows in between bright light spots unless you're a fan of film noir motif. 

Google the fixture manufacturer they should have a specification sheet. That sheet should have the fixtures photometrics. Then you can determine the footcandle level of one fixture. You're looking for 30 footcandles at task level (about 3' above the floor). Then look at the spacing of the fixture and determine the spread of the fixture and set your spacing that way.

But you can always do what I do. I start by Googling and contacting their tech help center or if they have a local manufacturer's rep I'll call and ask them to give you the spacing. If you give them the right information they can email you a rough plan and give you the right spacing and lamp.