None of this is correct, for QB and Job cost tracking and reporting.

You want to use Two Sided items, not that Expenses tab. Examples:

Other Charge Items for Waste Removal, Building Permits, and Equipment Rental. Two Sided = checkmark the box on the left, link the Left to Expense and the right to Income. Two Sided Service item, such as Electrician Sub and Plumber Sub. Two Sided Noninventory item for Doors, Windows, Roofing, etc that you custom order for the job. Use the Items on the Items tab, use Qty, use them on Purchases and job track them. Optionally, this is Billable or not, depending on the terms for the job, such as Fixed Bid (direct costs not billable) or Time & Materials (is billable).

That way, you don't Micro-manage the Expenses tab as one expense account mapped to one income account. And now, when you use Items on estimates, and on Purchases, credit card charge, and sales, you see the info in Job reporting. Run Job Profitability Detail and see if yours reads as nicely as my attachment. And Est vs Actuals, these rely on the use of Items. Not that Expenses tab.

And the Reimbursement is Gross Income. Yes, it is All Income. Do not wash away expense with negative expense (income) or income with negative income (expense).

This is Wrong: "it should not be under temporary power or temporary facilities BUT reimbursable Expense account.  Is this correct?"

This is more Items, not more Accounts: "I believe the way the account  is currently set up is Temporary Power or temporary facilities."

That way, you see Gross Cost and Gross Revenue, and the job reporting and the P&L reporting will show the Net. Even the Item Profit reporting will show the net.

This also is wrong for your accounting: "

So basically I have to set up two additional accounts - Reimbursable Expenses from Customer ( which is an expense account)"

For instance, as you see in my side-by-side of the P&L related to the Job details (follow the matching Color), I have one expense account for Jobs, one for Subs labor. The Details are Items.

"When the customer sends us the reimbursement check is there a way to keep track of that check to where it is reimbursed income and doesn't go directly to income."

It is your Payment to process against their Charges. The check is just part of Banking.

"The reason for this question is because in a recent General Liability audit they wanted the company's income."

And your data has been "netting in the accounts" all this time, which is Wrong. Everything is Under-reported.

"Technically, the reimbursement check is just that a reimbursement for a bill we already paid for the customer and should be separated from the actually income of the company."

No, that is not what you are doing, for a Business. You track Gross  Revenue and Gross Expense and you let Reporting show these differences. Otherwise, you completely under-report the scale of this business.

Please see my attachment.