Later stage in general. The earlier the company, the less structure or resources available for the support and mentorship needed for new grads. Exceptions always exist though.
Really, all sizes. There are about 50 companies in the directory now that are hiring new grads. The median size is 10 people; there are a bunch in the 2-3 person range and a bunch in the 100+ person range.
If you fill out a Work at a Startup profile, we'll automatically show you a list in the directory with just the companies hiring new grads.
Based on the typical compensation and culture: you describe the kind of employee they target.
And that is in a nutshell exactly what all the comments in the previous thread distill down to.