That is not a typical way of connecting floors to the foundation in many areas of the country. Is it a manufactured or modular house? My guess is the rim joist (what we would call the board that runs perpendicular to the joists, but "band joists" are used in some regions to describe the perimeter joists) is the 2x8(?) stacked on top, not the board precariously propped up on a few blocks. That is a beam that would be sized based on the area of the room it's supporting, the load (prob. 40psf), and the span. We do not know that information, nor any specifics of the codes in that area. We do call the board that rests on top of a foundation wall a "sill plate" but this is not a foundation wall, but piers... I think I can agree on "sill beam"
Terminology notwithstanding, you don't replace existing with smaller unless it was engineered or calculated out because it was usually calculated and specified as such originally. Given the connection between the piers and the beams I'm going to guess this was not exactly done according to code in the first place. Unless you are in a totally unregulated area of the country that's immune from all building codes (not sure if that even exists), that rim/beam/pier configuration would not be permissable at all. I'd definitely like to see some straps connecting at least the floor (rim joist & joists) to the beams, and something to connect the beam to the piers -- which would have to be drilled and epoxied (called "post-install" because it was done after the concrete was poured). I'd be pretty afraid to be near the house in a hurricane, twister, earthquake, dust storm, strong sneeze, or whatever typically hits your part of the world
I don't believe that pier foundations are allowed under load bearing walls... which one of those two sides certainly is (whichever side the roof starts on). They must be continuously supported unless an engineer has stamped their approval with an alternative. In either case there are minimum anchor bolt requirements in all buildings since the late 70's (given the sonatubes, I'd guess this is newer than that). Probably along the lines of 1 every 6' and within 12" of ends. So... that's for sill plates and continuous footings. Beams are usually connected via a metal connector appropriately sized for uplift and lateral wind loads. I've never seen the connector that is ok with ~5-1/2" space above the piers though! That's quite a lever arm that is set up.
But this is all a digression... The original question can be boiled down to:
Probably not. As I said, they would have had to reengineer the building, which I'm gonna guess they didn't do (why would they?). It's good they used PT, which the rest of the structure appears to be lacking. Technically only the wood that touches the cement or other moisture conditions that requires it, and that would be the "shim" material. I can't tell if that's PT or not. Cedar is not treated. BUT... that siding and belly band needs to be repaired or you will find that the kitchen or whatever that is sagging a bit more every year as the shims rot out underneath it.
Also that half-lap joint is interesting and probably not ok. It looks nice, but I see two problems. 1) It looks like there is a crack started in the existing beam because of the way the new was was propped up underneath it. 2) You have reduced the new beam to only a couple inches because that's all that's bearing! That is not a permissable notch. The new and old should just butt together over the support. Put a strap on it if you need them to be connected together.
- Is it ok that the corners don't meet?
Probably. It's not the most professional result and there very well may be minimum nailing requirements (usually through a metal connector) to connect the beams, but the #1 thing is about the bearing on the pier and both pieces have plenty of bearing.
I'd also guess there is going to be a deck or something attached there? I'm not sure there is much I'd consider ready to attach a ledger to. There's plenty of wood, but I'm not sure it's attached to much to keep it from falling off/pulling off the building!
My $0.02 says to go with your intuition and try to get this fixed. If there is a building department in your city/county/region, you can usually request an inspector come out and take a look, but that could very well get the contractor in hot water, so maybe talk to them first to see if they can answer to your satisfaction or remedy it. They may have sent over someone inexperienced or some other excuse and would fix it. They also might have thought they could get away using some piece of wood they found on the ground...
Good luck,
- Jared