The reason this doesn't work is because Minecraft's solid blocks like stone, wood, etc. are meant to take up the entire block space, they're supposed to cover any adjacent faces to them (in other words, the game won't render faces that solid blocks are directly next to, to allow for faster rendering). Unfortunately you cannot fix this, but there are some blocks that allow you to make into any 3D model you want. Cauldrons aren't solid blocks, they're under the category of "transparent", blocks that can be "seen through" (examples would be glass, glowstone (I think), and leaves). That explains why MrCrayFish's model works. but if you want to prevent the rendering problems, you have to cover the adjacent faces to solid blocks, which, unfortunately, limits the models you can make. Sorry if this inconveniences you but that's the sad truth. I hope sometime in the future Mojang can find a way, a way that doesn't make the game harder to run, to allow solid blocks to have a see through part to them. But as of right now, that's impossible.
Hope this helps you. And have a nice day/evening!! ^-^
And this is in no way an error with MrCrayFish's program, it's awesome, simple, easy to use, not so helpful for HD texture packs, but nonetheless, it's good, and I use it. Plus, it's free!! ^-^
Modeling does not affect lighting. You cannot make a diamond ore give out light level 16. This is the case with your logs. In Minecraft's code, that area of land is underneath a opaque block, and this cannot be changed without mods. Its the same if you make glass solid in your resource pack, it will still pass light out of it.
EDIT: In your case, just tick ambient occlusion an i think that will work c;