My question: Should the total of the balances in equity accounts associated with a particular bank account always equal the current balance of the bank account?
There is nothing in the program that associates an equity account with a particular asset accounts. So this is not enforced in the program.
In addition to the unrestricted equity account, 10 donor-restricted equity accounts are associated with the General Checking account (01-1110). If I run a tally of the 11 equity account balances at any given time, should that total equal the current balance in 01-1110?
The general accounting equation is:
Total Assets - Total Liabilities = Total Equity
Your post says the church has no liabilities, if there are no other assets other than cash accounts, then total cash would equal total equity. If you have any other assets other than cash accounts, then they might not equal.
Contributions toward three donor-restricted memorials (each with its own income/release/equity and expense accounts) are deposited in General Checking, then transferred to the Restricted Funds bank account (01-1230). Assuming I keep up with the transfers as the deposits are made, should the total of the three equity accounts always be the same as the current balance in 01-1230?
If the beginning balance of the equity accounts started off correctly, then after the transfer is made from the General Checking to the Restriced Checking account, it sounds like the balances should be the same.
The Building Fund bank account (01-1220) has three accounts associated with it. Again, should the total of the equity balances associated with those three accounts match the current balance of 01-1220 at all times?
If all the money for the Building fund is in the 01-1220 account then yes, the total of the equities for the building fund should match the total in the 01-1220 account. the same as above, if building fund money is deposited in the 01-1110 account, it will not match until the money has been transfered into the 01-1220 account.
It might be good to walk through a transaction so you can see how the money moves. Say you receive a $100 for the building fund. Here would be the transaction to record the receipt.
01-1110 General Checking $100 (db) (increase)
01-4510 Building Fund Income $100 (cr) (increase)
What has happened is General Checking is increased by $100. The income account shows $100, if we ran a balance sheet at this point, the building fund equity account would be increased by $100 because the income account was increased.
Move the money from the General Checking to the Building Fund Checking account:
01-1220 Building Fund Checking $100 (db) (increase)
01-1110 General Checking $100 (cr) (decrease)
This transaction deceases the General Checking and increases the Building Fund Checking. The balance of the equity is not changed. To change the balance of an equity account, we must use an income, expense, transfer or equity account in the transaction.
). We are using PC+ v10. 
I hadn't really thought about it.