To do it right, you should definitely remove the threshold and put in a waterproof membrane like ice and water shield (the peel and stick stuff that goes on the roof under shingles). You might be able to get it out by running a reciprocating saw under it to cut the fasteners, but from the pictures it looks like the entire door may have to come out. It should go under the threshold, down the vertical face, horizontally on the foundation and then down again at least an inch. Turn out the 2 sides so nothing runs behind the adjacent brick. Then you can patch the brick back in, or use some pressure treated wood if it will all be hidden by the walk. Put a couple of generous beads of caulk under the threshold.
There's really nothing inherently wrong with running a concrete walk right up to the sill if it's pitched away from the house and sealed with some waterproofing flashing. The concrete should not be poured directly against the wood unless it's pressure treated.
