Collection and return of homework and exams
To my mind, the biggest problem, after actually grading so much work, is the possibility of a student claiming that work was turned in but later lost by you or the TA.  You allude to this in your question.

You don't say where you are, but it is likely your university has a "learning management system," such as Blackboard, Brightspace, or Moodle,that can be used to collect take-home work.  For multiple-choice assessments, the learning management system can probably grade them, too.  The huge advantage of this is that there's no possibility of "lost" work; either the student submits the work or not.

For in-class exams, have a letter tray similar to the "inbox" one might have on a desk.  Require each student to deposit his own exam into the letter tray and, at the end of class, put a binder clip on the stack.  (For 100 students, you'll probably need four stacks of 25.)  Grade the exams on a completely clear desk and there's no possibility of lost work there, either.  If a Scantron machine is available and multiple-choice exams will work for you, use that for grading.  I have smaller classes, and I take each paper directly from the student's hand and mark the student's name on the class list.  Students observe me to do this, and I don't get complaints that I "lost" their work.

Work graded through the learning management system is returned electronically.  If you use Scantron forms for in-class testing, it may be possible to get the results electronically and return them via the LMS.  Your testing center can probably tell whether that's possible.  I have returned in-class paper tests through the LMS by scanning and uploading them.  We have a copier that can serve as a scanner, and that work can be performed by a TA or student assistant.  If you must return paper exams in class, have them sorted alphabetically and return them at the end of class so that students may leave when they've picked up their graded work.