How do I keep track of what event a particular member paid for when they make a deposit?

Deposits to member accounts are not linked to specific charges.   When you charge a member for attending an event, it reduces the value of his account by that amount.   When the person makes a payment, that deposit increases his balance.

An individual member account is like a credit card account.   When you make a payment to Visa, you don't pay for a specific item on your bill;  your payment simply reduces the total amount you owe on that credit card.   In the same way, payments from your members do not get posted directly to an event, but simply increase the funds available to that member (or reduce the amount he owes the group.)

We do provide the option of noting the event for which a payment was intended.  This does not post affect the balance of the event, but serves as a "memo" to indicate the intent to pay toward that event.

But my group requires that all members pay for an event before the event occurs. How do I handle that?

If you require members to pay for each event in advance, create a Charge Member Account For Event transaction for each member who plans to attend.   This will reduce each member's balance by the cost of the event.  Payments can then be received and deposited into each member's account.   If you then view Individual Balances or Report Account Balances, you can see which members have negative balances and thus have not yet paid for the occasion.

You can also view the current account balances for each participant in an event from the Calendar by configuring your User Roles to be authorized for the "View Participant Individual Balances on Event" task.

As noted above, you can also configure transaction types to allow deposits to reference the event for which they are intended. This does not affect the balance of the event, but these payments can then be viewed when you display the participants from the Troop Calendar.  If you use this feature, you will need to set a policy that determines whether a member should be allowed to attend an event if they have a negative balance in their account (i.e., they owe the troop money) but made a payment intended for that specific event.

How do we use the system to keep track of group assets, like the value of merchandise we are selling for a fundraiser?

Most groups do not require this type of accounting.  However, it is possible to use the fund accounting feature to set up various asset accounts.   While we do not encourage this, with enough creativity you may be able to define the necessary categories, funds, and transaction types to make this work.