What screws to use to attached tie-downs from shed roof rafters to top sill

1. Measure the 2 diagonal distances to the corners. If the measurements are not the same, the floor is not square.

2. Not sure what you are levelling to. A level, framing square, and careful measuring solves most layout problems.

3. 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.

4. 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.

5. Block between joists at all supports. This is a typical code requirement.

A ridge board isn't a "beam", per se. It's often just a nice way to bring the rafters together. A proper beam would have a supporting post at each end.

You don't need a ridge beam, but you do need a way to keep the walls from spreading. This can be accomplished with "collar ties", boards spanning the shed at ceiling height, or with large gussets further up the rafter slope. One collar tie would be enough in your case.

I built my 12x10 shed by using 4' by ~18" OSB gussets at the peak of each 2x4 rafter set, essentially resulting in a truss, which proved very solid:

Source: http://halbergengineering.blogspot.com/2010/10/structural-engineering-for-minnesota.html