I would recommend you hire a structural engineer to review the contractor�s plans. Don't understand your comment on the contractor cutting the joists, will he sister new joists to the old joists. Are their local permits in place for this renovation? Hard to comment on using the chimney cinderblocks since its support structure may not have been evaluated, the structural engineer can answer that question. If this part of the renovation is not done correctly there will be major issue later so have it evaluated by a professional engineer.

He's cutting the joists since there will be joist memory, so he believes cutting them will remove downward pressure between the steel beams and joists. I know the contractor has been talking to a structural engineer, but it sounds like I should have him come out to evaluate things in person. Thanks!

I would have the enginner sketch out the procedure and show the existing framing with the modifications superimposed so that the whole process (including the construction sequence) is documented. The engineer should also sign and seal the drawings.

Is the local building official involved?

The contractor is leaving involving the inspector up to me. He thinks the inspector is worthless because he doesn't really know. He says we could involved a structural engineer, but it will cost $500 to have him come out and put his seal on the drawings. He guarantees the engineer will agree with him. He's confident of his plan and he's accepting all liability (in writing) for what he proposes. He has insurance and he's been great to work with. I trust him, but the scope of this makes me nervous.

Warranties are only good if the contractor is still in business. Structural problems will not show immediately they will start to show as cracks, unlevel surfaces, etc. Also as a structural element there could be a structural collapse. The $500 is cheap relative to the cost of a redo. Have the engineer come out, document and sign/seal as suggested earlier. I would also suggest you get local inspectors looking at the work. Where I live if you don't get permits and it�s noticed you will be asked to pull the permits with the inspector having the right to do a tear down to inspect.

One other point it's never a good idea to cut the joints since it lessens its weight load capacity and may cause bounce. If it�s done then the joist needs to be sistered to add back the strength. Get the engineer out!!

Below is a web page that described sistering.

http://www.familyhandyman.com/DIY-Projects/Flooring/Floor-Repair/how-to-make-structural-repairs-by-sistering-floor-joists

I've done similar jobs on houses but have always gotten engineers involved, permits or not. Normally my rule of thumb is if it was properly done to begin with than fix what's broke. If it was half-a$$ to start with get an engineer and make sure your loads are properly supported.

What is going to support the steel beams?

You are creating point loads at their ends.
Is the support going to be adequate?

Cutting the joists back is normal when putting steel into the floor, to allow room for the steel. After the steel is in you sandwich it with 2x's bolted through and reattach joist to them. As brick said load points are the biggest concern. Engineer is a must, I personally don't put much faith in inspectors I've seen way to much get by them, and they use engineer's drawings to make sure it's done right