Integrated Security=True;
So it's connecting as the windows user you're logged in as.
What  permissions does that login have on the SQL Express instance?

You should be getting an error, yes. It's not a fatal error though, it doesn't abort the batch or the session, so without error handling you might not notice you're getting an error. Without sufficient permissions, the function doesn't return useful values though, and hence the code inside the IF blocks won't execute.
https://docs.microsoft.com/en-us/sql/relational-databases/system-dynamic-management-views/sys-dm-tran-active-transactions-transact-sql

Yes, the stored procs will execute individually, there's nothing wrong with the individual stored procedures. The problem is that udf_TransactionExists() does not check whether that session has an open transaction. It checks the server-level DMV to see whether there are *any* transactions named 'Add' or 'Edit' at all, from any of the running sessions, and without sufficient permission, won't return what's expected. same with the other two functions that check for transactions

No, they don't.

The first executed the insert code if that user has added a record and the update code if the user has edited a record

The second checks to see whether anyone using the server has a transaction named add, and if so it inserts a record. If no one on the server has a transaction named add, and anyone on the server has a transaction named edit, then it updates the row. If it has insufficient permissions to check that, it does nothing at all.

The debugger's a right bloody pain, and requires sysadmin access which means it seldom gets used. More normally we use PRINT to debug, or insert into logging tables or stuff like that. PRINT requires that you're running from Management Studio, or are looking for the messages being returned from the server (and I have no idea how to do that in VB)

You can take what I said (here and last time you asked about this) as an accusation, or you can take it as things to improve. Your choice.