Professor not giving solutions to exams and assignments that I need to grade

I can't tell quite what your concern is, so I'll address both.

Is it OK for a non-professor to write solutions and design a rubric?

In my experience (large US universities, STEM), this is extremely common. I disagree a bit with others saying that only professors can make such determinations (though of course the instructor of record can make such determinations if they are inclined to do so).

Is this abusive toward the TA?

How are you being paid?

• If it is hourly, then there is no issue; you should solve the problems and make a rubric however you see fit, and then bill for that time as normal.
• If it is a set stipend, then there is usually a nominal time requirement (in my university it was 20 hrs/week for grad students). If making solutions will put you way over this, it's fine to raise the issue with the professor and/or with the administration (in fact, you may technically be required to do so)

One other tip -- you might consider taking the top-performing students' solutions and using them as a starting place for your own solutions. Of course, you'll have to be sure you're grading those students fairly and catching any errors. But I've found that the I can often create a solution set just by mixing and matching solutions from the top ~3 students and adding a few expository details.

Ask to be paid for all the work you are required to do.

Most graders are employed on casual rates and paid per hour. If that is the case then you should speak to the administrator in charge of employing you and confirm with him/her that you will have to find the solutions to the assessment problems and determine an appropriate marking scheme (for partial marks) as part of your grading duties, and that this will take you extra time. Make it clear that you expect to be paid for the additional time you spend on the work, and get confirmation that this will occur before you start working.

If you are usually paid a flat-amount for grading, or an amount per script, and this usually includes being supplied with solutions, then you should again raise this matter with the administrator that is in charge of employing graders, and get confirmation that you will be paid extra for the additional time you spend finding solutions to the assessments and creating an appropriate marking scheme. Again, make it clear that you expect to be paid for the additional time you spend on the work, and get confirmation that this will occur before you start working .