Framing a 10x12 wood floor, using 2x6 PT lumber
Measure the 2 diagonal distances to the corners. If the measurements are not the same, the floor is not square.
Not sure what you are levelling to. A level, framing square, and careful measuring solves most layout problems.
16" spacing is fine as long as your sheathing is rated for 16" spans. Sounds like they are spanning about 6 feet? 16d common nails (3-1/2") are usually used for framing. I'm concerned with spanning 10 feet with 4x4s supporting 3 feet of contributing floor load. Plus wall dead weight and maybe supporting the roof? I'm pretty sure if you checked a load table, this is inadequate.
Hangers are not needed since the joists sit on top of the 4x4s. Tie downs from joist to 4x4 is a really good idea. 32 inch spacing along each 4x4 should do it.
Block between joists at all supports. This is a typical code requirement.