Can QBS SE provide running transaction balances (account register)?
QBSE does not have an account register at the moment, goceano18.
What it offers is simple tracking of business income and expenses into the correct Schedule C categories, mileage tracking, and estimated taxes calculation. You can take a look at this article to get more details how the program works: QuickBooks Self-Employed Overview.
You'll want to switch to the QuickBooks Online version which offers account register balances an a fully functional Chart of Accounts and account registers.