At the end of a given period should your checking account balance match the amount on your trial balance sheet?

Yes. But keep in mind that you would need to do a bank reconciliation of your account before assuming that's a correct balance. The book account (and therefore trial balance) amount should be correctly recorded as to what is really in your bank account. If there's differences, you'd have to discover what they are and get them taken care of. Since the checking account is the first place of entry, you would be entering that into your books from the check register, and it should all be the same things.

Your balance per your trial balance will not always match the amount on your bank statement. One example of why this could happen is outstanding checks. If you write checks on the last day of the month you credit cash on your trial balance for that amount, but obviously they would not show up on your bank statement because they have not yet cleared the bank. This is why you reconcile your trial balance cash balance to the balance per your bank statement.