@LibbyVB 

Only reason I can think of that QBO won't allow you to merge the accounts is if they are different types.  So, check they are both bank type accounts - checking or whatever first.   If they aren't the same, then you will need to edit that before you can attempt to merge.   BEFORE you attempt to merge them, check there are no duplicate entries and either delete or exclude them from the register.

Assuming they are both the same type of account (or you can edit them to be the same), then edit and rename one of these accounts.  Usually, I copy the name of the account that is connected at the bank and make the other account name the same to merge them. Once you do that and hit 'save', QBO should ask you about merging them.  Say yes and then save.  Double check that you don't have duplicate entries and then reconcile.

As for the rogue JE...   sounds like the adjustment was made for you at the end of reconciling?  the system suggests an entry to force reconcile when you hit the finish now button during reconciliation.  I'm not aware of an automatically adjusting JE to reconcile the account.

When I have clients in this situation, it's very tedious but often worthwhile starting back at a logical starting point like Jan 1 and undoing all the reconciliations and starting over to make sure you have all the entries that the bank has.   Check if you are using bill pay that you don't have duplicates with the bank feed.   If you do, delete the one that does NOT have the green boxes that denote downloaded from the bank.  If you delete the ones downloaded from the bank from your register, it will pop back up in the bank feed to be categorized again.   Another frequent culprit of duplication is credit card payment entries, so look for those.  Also payroll if you are running that can sometimes not tie out properly with the bank feed if the account name you have in the accounting section is not correct.  

Let us know how you are doing!

Plum