Dear TG,

the 4x6 will work if the plate is fully supported, say you are building of a concrete slab. If you are using piers then you might consider something a bit heavier.
If you end up using the 4x6 then you just might have to put the floor joists on top of the plate. Although if you have a slab that will not be necessary.
The lap joint will work just fine and is easy enough to cut. If you use a through tenon on your post that will also serve to secure the lap joint. There is also not as much ( almost none) tension on this joint anyway. Not compared to a tie beam.

Regarding the dimension and if one could call it a beam. It certainly is not but what you are going to be building is still a timber frame. So no worries there about stick/beam sizing. Many/all braces used in timber frames are not actually a beam/timber so why worry about a smaller sill plate?

Edited for additional info