The 10′ span looks like a reasonable span for 2×10 floor joist. They can span up to about 15′. The spacing is typically 16″, not 10″. If 2×12 floor joist are used instead, they can span up to about 18′.
In general, a structural engineer should provide you with calcs for these joist and beams. Depending on it’s span, the double 2×10 beam may not work. In addition, the stair opening isn’t framed out correctly, and addition beams are required