Thank you for using Just Answer. I will try to answer all your questions, but a number of answers may depend on circumstances you have not specified.
Your first question: It would generally be better for you if you allocate all the expenses to your business (Schedule C), but the law and regulations suggest you should allocate expenses which are appropriate for either your employment or your business, proportionally to gross income. In other words, if you gross $30K from your business and $45K as a professor, you should allocate 3/5 of your expenses on form 2106 (or 2106-EZ) and 2/5 of your expenses on Schedule C (or C-EZ).
Ok, that makes sense. Any insights on the second part of the question?
Claiming travel expenses as childcare is more complicated. I would say you can only claim those expenses as childcare expenses; your friend's travel expenses and meals (but not your child's) should be allowable as childcare.
I'm not familiar with the nuances of Georgia law, but it's possible that, if you have employees and make child care available to those employees, you might be able to deduct your child care as a business expense. This wouldn't work for your Federal taxes unless you set up a qualified plan in your business, and probably not even then. I'm not an expert in drafting qualified benefit plans, so you would need to find another expert to answer whether it could be done at all; I'm just saying it's possible.
That covers everything I can think of at the moment; if you need further clarification or details, please let me know.
I can see that you typed something, but I can't see what it is, at the moment. Please bear with me.
Sorry, I've now switched browsers, and it seems to work. (I'm not a computer expert.) Any more questions?