"and showing me with a higher income than I am actually receiving, or that the government is taxing me and also taxing my subcontractor."
You are confusing Gross and Net.
This is Bad guidance: "What happens in the profit and loss is the expense goes to that specific account, but so does the income (since it is marked as billable, it is treated as a reimbursement). SO, if all these kinds of expenses are billed out, then your profit and loss statement should show a ZERO for expenses in this account.."
That is not what you do, for tax reporting or business tracking. Gross = In Full. You Track everything. I use this example in class:
My son sells lemonade and makes $2.
I build and sell robots. I sell One Robot this year for $1 million. I spend $999,998 to build it on labor and materials. I do NOT want to report $2 only, as income. My son and I have completely different Business activities and the Numbers prove this.
If you had a $1 Million year Gross, you should be proud of that. You report Gross or Total income, and Gross or Total Expense. You don't try to Hide Sales for purposes of avoiding tax reporting. Any agency with authority can audit your records and see the money flowing.
The "$2 Net" if you dump negatives and positives into one account, is called Netting the accounts. That is Not what "reimbursement" means. It does not Remove the fact of your expenses. Reimbursement for business = Sales. You just didn't mark it up, and won't see a Profit (net) when you look at the P&L = income over expense.
"Do I need to open up an S-corp to avoid this?"
No, because this Does Not Change; it still is Gross Vs Net. You track everything in full = gross.
"Where do I denote my expenses like this in my tax forms?"
In the Expenses section. Get your tax form, such as Sched C. It's right there.
"Am I required to submit to my subcontractor a 1099 at the end of the year?"
The 1099-Misc rule = "Paying $600 or more." You give them a W9 before giving them the check that takes you to $600 or more paid this year to them.
The penalty for this is $100 per day per form not filed.