We use a lot of LLCs for our real estate business, which is obviously not similar, but can provide some guidance on your questions. This is not tax or legal advice on what is the best strategy. This does not sound like the kind of profession that would need liability protection.
If you setup a single-member LLC, it would not require a separate EIN since its disregarded for tax purposes anyway and everything would flow thru to Schedule C as a sole proprietor. On this form you could deduct all of your business expenses.
The colleges would then need to make payments to this corporation and you would have a bank account specifically for this entity. You CANNOT mix personal and business transactions, so the college cannot write a check in a personal name and then deposit into the business. The colleges would still need to issue 1099s in the name of the entity come tax time.
If the colleges don’t want to issue 1099s in a person’s name then I doubt they would do it in a corporations name either. Best to find out up front before going through the trouble.
I looked into this for my self employed wife and consulted with a professional. Conclusion was that it’s not worth the trouble and expense unless you make a decent amount of money (over $100K but there are differences of opinion on this). The reason to set this up is to defer taxes and only pay tax on the “salary” you pay yourself from the corp.
Since your friend is being paid as a W-2 employee the employer is paying the other half of social security and medicare tax. So even less reason to set up an S-corp.