Can I Move Interior Load-Bearing Walls Without Changes/Additions to Beams/Piers/Foundation?
Wow this is complicated...I love it. 

You say, you want to know “if I can make some changes to my house's interior walls without making changes or additions to the underlying beam/pier/foundation structure.” Actually, it’s way more complex than just that...

We design from the top down and build from the bottom up. I’d like to start with your roof and how those loads are transferred down through your house. From your picture, it appears that ALL loads transfer to the exterior walls. That is to say, no roof joists or supports rest on ANY interior walls. That’s good. That allows you to move interior walls without affecting roof loads transferring down the walls to the interior foundation system. 

Floor framing is designed for the critical span (load) and then used throughout the first floor. The critical span appears to be in the triple door entry area (I presume it’s the Dining Room) at about 8’. (Most of the spans are 6’ ) 2x6 joists at 16” on center spanning 8’ can support about 72 psf (old size lumber dimensions) depending on the species and grade. This is substantially above what is required by Code. (Code requires 40 psf “live load” plus an estimated 15 psf “dead load”. )

The 4x4 floor beams span about 5’ (4x4 posts are about 5’ on center) and can support about 250 - 400 plf depending on the species and grade. However, the total floor load on the 4x4 beams is  about 385 plf. (Half of one 8’ span plus half of one 6’ span resting on the joists supporting 55 psf = 385 plf.) Therefore, the 4x4’s are barely adequate, (unless there is a roof load on them.)

Next is footings. No additional weight (load) is being transferred to the footings by moving the walls, so I don’t anticipate any problems with the footings. 

Summary:

So, because the roof load is on the exterior walls only, (you should verify that I’m correct by looking up in the attic and checking the entire attic...your pic is for only about 1/3 of attic), then moving walls DO NOT affect your foundation system. 

However, there are some major walls that appear to support some roof loads that appear to rest on the floor joists. So, to answer your question about changing your foundation, we won’t know until we know about your roof. 

Area A: I think you are correct, the wall is non- load bearing. Lath and plaster are not shear walls. I wouldn’t worry about removing 3’ or so of wall. 

Areas B, C, D, and E: Again, I think you are correct about increasing the size of ceiling joists or providing additional support where the ceiling joists do not lap at the new wall locations. However, when you remove the walls that are perpendicular to the exterior walls, you need to “splice” the exterior double top plate. Currently, the walls are “woven” into the exterior walls. Cutting them off requires you to splice the double top plate with 1) tying the roof joists to the double top plate (I like Simpson H-1 clips) and 2) Installing a strap over the top plate where the walls were woven together (I like Simpson MSTA or LSTI in 48” long lengths.)