When the church pays for a service, money leaves the organization from an asset account which indirectly lowers the equity (or fund balance).

The work around for Quickbooks and funds is to use classes. However, according to Quickbooks own site they say this about classes.

The Balance Sheet by Class report can’t classify some transactions in QuickBooks. Some of those transactions create unclassified amounts on the report.

By not classifying every single transaction's disbursements, your balance sheet will never be right. Some of the transactions that can't be classified are journal entries with unbalanced classes, a payroll liability check, and transfer of funds between two church funds.

This is an accounting 101 question. Please see Investopedia. This equation is not the same as Assets - Liabilities = Equity (or Fund's Balance). The equation has multiple formulations based on what is needed - ie balance sheet.

Donation funds and accounting funds are different. You will want to review this article. Donation funds are reported on differently than accounting funds for example. Donation funds start over each year. Accounting funds carry a balance forward each year. This is just one difference.

To fully understand this question the user must understand the fund accounting principles. They would want to review the What is Fund Accounting article. Simply stated using revenue accounts has too many problems to figure out how much each church function (ie mission) has to spend or has spent.

When multiple checkbooks (or worst sub accounts of checkbooks) are used, that means the system can't do proper fund accounting. It can't take the assets account (the checkbook) and break it up into various funds - or equity pieces. Using any type of accounts in the chart of accounts to define the funds is incorrect. They belong in the equity section. To read more, check this article out.

The other options while they are paid out of the church's accounting checkbook, they are not considered expenses. Paying a bank loan lowers the principle of a liability.

The only person that can restrict contributions are donors. While the church's council may say they are taking donations for a purpose, they can later change that purpose by another church's council vote or whatever procedure they have described in their by laws.

GAAP stands for Generally Accepted Accounting Principles. See the definition at the CFA Institute. Keep in mind all churches must follow GAAP.

Liabilities plus equity will equal assets. This comes down to the balance sheet equation where Liabilities + equity equals assets.

The definition of fund accounting that is used in non profits is the following:
"Fund accounting is an accounting system for recording resources whose use has been limited by the donor, grant authority, governing agency, or other individuals or organizations or by law.[1][2] It emphasizes accountability rather than profitability, and is used by Nonprofit organizations and by governments. In this method, a fund consists of a self-balancing set of accounts and each are reported as either unrestricted, temporarily restricted or permanently restricted based on the provider-imposed restrictions."

There are two main accounting methods called cash accounting and accrual. See Investopedia.

Contributions (donations) is revenue to the church.

According to FAS Topic 958 the Statement of Activities is equivalent to the income statement that for profit industries use.